c 2016 Odin Roth Jesse

Algebraic Characterization of Non-negativity of Polynomials over Polytopes

by

Odin Jesse, B.S.

A Dissertation

In

Mathematics and Statistics

Submitted to the Graduate Facultyof Texas Tech University in

Partial Fulfillment ofThe Requirements for the Degree of

Doctor of Philosophy in Mathematics

Lourdes JuanChair

Ismael de FariasCo-chair

Victoria Howle

Arne Ledet

Kevin Long

Mark A. SheridanDean of the Graduate School

May 2016

c©2016 Odin Roth Jesse

Texas Tech University, Odin Jesse, May 2016

Acknowledgments

I would like to express my sincere gratitude to Professor Lourdes Juan and Professor

Ismael de Farias for their guidance and patience throughout the course of this project.

I am also very grateful to Dr. Arne Ledet, Dr. Victoria Howle, and Dr. Kevin Long

for their insight and assistance.

I would like to thank my friends and colleagues Scott and Erin Williams, Fatih

Koksal, Lars Christensen, Anton Kleiwer, Amin Nikakhtar, and Yang Yu for making

the Texas Tech math department such a lively place to work. I am also grateful

to Dr. Carl Seaquist and Dr. Mara Neusel for their enthusiasm and encouragement

during my interview for graduate school. I would also like to extend my appreciation

to Gene Gray for his help navigating the bureaucracy involved in graduation.

I am indebted to Jens, Luke, Kent, Lauren, Brett, Dane, Kira, Peter, and Sten

Langsjoen for making sure I didn’t take the romanticism of academia for granted.

Finally, I would like to thank my parents Richard and Kristen, my sisters MK and

Jenna, and my partner Megan for their love and support.

ii


Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Fundamental Tools and Concepts . . . . . . . . . . . . . . . . . . . . 4

2.1 Computing Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Polynomial Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Real Radical Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Existing Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Theta Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Polyhedra in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Geometry of (1, 1)-SOS Ideals . . . . . . . . . . . . . . . . . . . . . . 25

4 Preliminary Observations and Technical Lemmas . . . . . . . . . . 29

4.1 Isometries of the Cube . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Idempotence of Affine Polynomials . . . . . . . . . . . . . . . . . . . 32

4.3 Constructing the Ideal of a Binary Polytope . . . . . . . . . . . . . . 34

5 Algebraic Characterization of (1, 1)-SOS Ideals . . . . . . . . . . . . 37

iii


5.1 Distinguished Facets and Generators . . . . . . . . . . . . . . . . . . 37

5.2 Full-dimensional Polytopes . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 The Facet-Generator Correspondence . . . . . . . . . . . . . . . . . . 45

5.4 Sums of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6 Necessary Conditions via IP Feasibility . . . . . . . . . . . . . . . . . 64

6.1 Hadamard Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Integer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Some Important Counter-Examples . . . . . . . . . . . . . . . . . . . 75

7.1 Nuances of Facets and Generators . . . . . . . . . . . . . . . . . . . . 75

7.2 Projections of Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 79

8 Comparison of Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.1 Computational Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.2 Data and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.3 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

iv


Abstract

In this work we provide a system of algebraic techniques for recognizing polynomial

ideals with structures that are conducive to the study of sum of squares (SOS) poly-

nomials. Ideals with these SOS properties correspond to regions in Rn over which one

may quickly and efficiently obtain certificates of non-negativity for polynomials. As

such, these ideals are exceptionally interesting in the context of polynomial optimiza-

tion. A geometric method of recognizing the regions in Rn with this cooperative SOS

feature was developed in 2011. Our work involves a series of sufficient conditions and

necessary conditions designed to recognize the ideals corresponding to such regions

based solely upon their algebraic structure. This will allow researchers to identify

ideals that are conducive to SOS techniques without the need to perform expensive

geometric computations. Our work concludes with a comparison of algebraic and ge-

ometric efforts to recognize SOS ideals in terms of running time and accuracy. These

results show that algebraic techniques run exceptionally quickly but at the risk of

inconclusive results. This risk of failure can be substantially reduced if our focus

is restricted to ideals corresponding to polytopes with a relatively large number of

vertices.

v


List of Tables

8.1 Average Running Times for Low Dimensions (seconds) . . . . . . . . 86

8.2 Average Running Times for Large Polytopes in R6 and R7 (seconds) . 88

8.3 Summary of Average Running Times for Low Dimension (seconds) . . 88

8.4 Summary of Average Running Times for Large Polytopes (seconds) . 89

vi


List of Figures

3.1 The polytope for a (1, 1)-SOS ideal . . . . . . . . . . . . . . . . . . . 26

3.2 The polytope of an ideal that is not (1, 1)-SOS . . . . . . . . . . . . . 27

4.1 Isometric polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Using elimination polynomials . . . . . . . . . . . . . . . . . . . . . . 35

5.1 The polytope for an ideal with no quadratic distinguished generators 40

5.2 A polytope with bivariate facets . . . . . . . . . . . . . . . . . . . . . 46

5.3 A polytope without bivariate facets . . . . . . . . . . . . . . . . . . . 48

5.4 The polytope corresponding to I . . . . . . . . . . . . . . . . . . . . 54

5.5 Two binary polytopes in R3 . . . . . . . . . . . . . . . . . . . . . . . 60

5.6 Sums of Ideals vs Intersections of Polytopes . . . . . . . . . . . . . . 61

6.1 An illustration of necessary conditions . . . . . . . . . . . . . . . . . 64

6.2 An illustration of necessary conditions (cont) . . . . . . . . . . . . . . 74

7.1 Nuances of the facet-generator correspondence . . . . . . . . . . . . . 77

vii


Chapter 1

Motivation

The interplay between algebraic geometry and optimization has a rich and interest-

ing history with a number of exciting recent developments. The tools of algebraic

geometry have proven instrumental in integer programming [9] and have been uti-

lized to provide valuable insight into the characteristics of feasible sets for convex

optimization problems [34]. More recently, work in this field has gravitated toward

the application of techniques from algebraic geometry to semidefinite programming

problems [25]. A central object in these studies is the concept of a sum of squares

polynomial.

Definition 1.0.1. Let f(x) = f(x1, . . . , xn) be a polynomial. We say that f is a

sum of squares (SOS) polynomial if f can be written in the form

f(x) = p21(x) + · · ·+ p2m(x)

for polynomials pi(x) ∈ R[x] for i = 1, . . . ,m. If for a positive integer k we have

deg(pi) ≤ k for all i = 1, . . . ,m we say that f is k-SOS.

We will make regular use of SOS polynomials. The following notation will be

helpful to us.

Notation: Let Σ[x] ⊆ R[x] denote the set of SOS polynomials in R[x]. Let Σk[x]

denote the set of k-SOS polynomials in R[x].

Notice that if f is SOS then f is non-negative over Rn. This simple observation

is exceptionally useful in a number of applications. Many optimization exercises can

be rewritten in terms of non-negativity over Rn. For example, the problem

minimize f(x)

1


such that x ∈ A ⊆ Rn

can be rewritten as

maximize t

such that f(x)− t is non-negative over A ⊆ Rn

We can address this non-negativity constraint using SOS polynomials. This can

be implemented naturally whenever our region A is a semi-algebraic set given by

polynomial inequalities

A = {x ∈ Rn | gi(x) ≥ 0 for i = 1, . . . , k}

where gi(x) ∈ R[x] for all i. Consider the comparison of the problem

maxx∈A{t | f(x)− t ≥ 0} (1.1)

with the problem

maxσ,t{t | f(x)− t = σ0(x) +

k∑i=1

σi(x)gi(x) where σi ∈ Σ[x] for i = 0, . . . , k} (1.2)

Certainly, any σ ∈ Σ[x] is non-negative over A ⊂ Rn and gi(x) ≥ 0 over A by con-

struction. Hence, the SOS expression for f(x) − t in formulation (1.2) is sufficient

to claim non-negativity over A. So (1.2) provides a natural relaxation of (1.1). This

technique was first explored by Putinar in [27] where a hierarchy of relaxations was

constructed based upon the maximum degree of the SOS polynomials σi in the repre-

sentation. For d ∈ N the dth value of the sequence is obtained by replacing Σ[x] with

Σd[x] in (1.2). The tools of algebraic geometry can be utilized to compute the values

in this sequence via semidefinite programming (SDP) algorithms. We will explore

this in detail in the following chapter.

The primary alternative to the SOS techniques presented in (1.2) utilizes the

Krivine-Stengle [33] certificate of non-negativity over A. This method assumes that

A is compact. Scaling as necessary, we may proceed under the assumption that

0 ≤ gi(x) ≤ 1 without loss of generality. We then approach the non-negativity in

(1.1) using the following construction. We let α = (α1, . . . , αk) and β = (β1, . . . , βk)

2


be k-tuples of positive integers and we write (α, β) ∈ N2k. For a positive integer d let

N2kd denote the set of (α, β) ∈ N2k that satisfy

∑ki=1 αi + βi ≤ d. Then the dth value

in the Krivine-Stengle hierarchy is given by

maxλ≥0,t

t | f(x)− t =∑

(α,β)∈N2kd

λαβ

(k∏i=1

gαii (x)(1− gi(x))βi

) (1.3)

where λ is a tuple of non-negative real numbers indexed by (α, β) ∈ N2kd . Since

λαβ ≥ 0 and 0 ≤ gi(x) ≤ 1 by assumption, we note that the representation in (1.3)

provides a certificate of non-negativity over A. This sequence of relaxations has been

extensively studied and several specialized variations have been employed to great

success in many optimization contexts [31].

The hierarchy constructed in (1.3) enjoys a significant luxury in that the values

in the sequence for d = 1, 2, . . . can be computed using LP solvers. These are very

thoroughly researched and optimized tools that can handle problems on a far greater

number of indeterminants than the SDP techniques required for (1.2). On the other

hand, it has been shown that the Krivine-Stengle methods cannot achieve an exact

solution for most convex optimization problems [14] (the convergence of (1.3) is only

asymptotic). Meanwhile, the SOS techniques in (1.2) converge to an exact solution

in finitely many steps for optimization problems over a wide variety of compact re-

gions, including all polytopes [10]. Jean Lasserre, a leading researcher in this field,

has recently proposed a hybrid approach to the study of non-negativity [13] that em-

ploys a SOS formulation in the spirit of (1.2) that is facilitated by features of the LP

formulation in (1.3). Lasserre’s modification of the SOS hierarchy excels over certain

compact regions in Rn with exact convergence occurring at the first step of the relax-

ation for select cases [15]. Our work herein aims to contribute to the understanding

of the regions over which SOS techniques such as these perform particularly well.

3


Chapter 2

Fundamental Tools and Concepts

The desirable cases for the SOS procedure in the previous chapter involve feasible

regions over which non-negativity of polynomials is “cooperative” with SOS repre-

sentations. We will make this notion formal momentarily, but we first introduce some

computational tools.

2.1 Computing Sums of Squares

A polynomial f(x) ∈ R[x] is a sum of squares if and only if f can be written in the

form

f(x) = uTQu

where u is a vector of monomials and Q is a positive semidefinite (PSD) matrix. We

illustrate with an example.

Example 2.1.1. The polynomial

f(x, y) = 20x4 + 52x3y + 35x2y2 + 4x2 + 6xy + 1

Can be written

f(x, y) =(

1 x2 xy) 1 2 3

2 20 26

3 26 35

1

x2

xy

4


Notice the matrix here is PSD with Cholesky factor 1 2 3

0 4 5

0 0 1

One can verify

f(x, y) = (1 + 2x2 + 3xy)2 + (4x2 + 5xy)2 + (xy)2

Notice that the monomials in the SOS representation of f are the monomials

present in the vector u. Additional information about the relationship between SOS

polynomials and PSD matrices can be found in [32].

Given a polynomial f(x) ∈ R[x], the corresponding PSD matrix Q may be com-

puted using semidefinite programming solvers. All such computations herein are

performed using the SeDuMi package for MatLab [30]. We now discuss how to use

SDP solvers to find the PSD matrix Q associated to a SOS polynomial f .

Notation: Given an n-tuple αi = (αi1, . . . , αin) ∈ Nn we denote by xα

ithe mono-

mial xαi1i x

αi2

2 · · ·xαinn ∈ R[x].

Given a polynomial f(x) ∈ R[x] we seek to find a SOS representation for f , if one

exists. We construct a vector of monomials

u =

xα

1

xα2

...

xαm

We then consider the equation

f(x) = uTQu

5


Regarding Q symbolically, Q = (qij), we may expand

p(x) = uTQu

to obtain a polynomial p(x) with coefficients that are linear polynomials in qij. By

comparing the coefficients of p(x) with the coefficients of f(x) we obtain a system of

constraints on the entries of Q

`1(q11, . . . , qmm) = a1

...

`k(q11, . . . , qmm) = ak

Hence, we seek a positive semidefinite matrix Q whose entries satisfy the con-

straints above. This is a SDP problem. The SeDuMi package, amongst others, can

provide the matrix we seek, if it exists.

Notice that in the procedure outlined above we must simply guess which mono-

mials populate the vector u. This, of course, is not a very viable technique for

application. We may eliminate this guesswork by using some tools from algebraic

geometry. We outline the approach below.

Definition 2.1.2. Let I ⊆ R[x] be an ideal with variety V(I) ⊆ Cn. By VR(I)

we denote the real variety of I given by

VR(I) = V(I) ∩ Rn

Suppose we have an ideal I ⊆ R[x] and a polynomial f(x) ∈ R[x]. Notice that f

is non-negative over VR(I) if we can write

f(x) ≡ p21(x) + · · ·+ p2m(x) mod I

6


To determine if such a representation exists we slightly modify the techniques outlined

above. We construct a vector of monomials

u =

xα

1

xα2

...

xαm

and we consider the symbolic expansion p(x) = uTQu. We study the congruence

f(x) ≡ p(x) mod I

Let fnf and pnf denote the normal form obtained by reduction of f and p respec-

tively by a Grobner basis for I. We may then replace the congruence above with

traditional equality

fnf = pnf

and follow the original procedure for finding Q. For additional details we recommend

M. Laurent’s manuscript [18] as well as P. Parrilo, R. Thomas and P. Blekherman’s

text [3].

Of course, we still encounter the problem of not knowing which monomials popu-

late the vector u. However, if we know ahead of time that f is k-SOS then we know

that we need not consider monomials of degree greater than k. This motivates the

following two definitions.

Definition 2.1.3. Let I ⊆ R[x] be an ideal and let f(x) ∈ R[x] be a polynomial.

If we can write

f(x) ≡ p21(x) + · · ·+ p2m(x) mod I

we say that f is SOS mod I. If deg(pi) ≤ k for all i we say that f is k-SOS mod

I.

Definition 2.1.4. Let I ⊆ R[x] be an ideal. If every polynomial f with deg(f) ≤ d

that is non-negative over VR(I) is k-SOS mod I we say that I is (d, k)-SOS.

7


So, if we know that the ideal I is (d, k)-SOS then the statement that a polynomial

f is non-negative over VR(I) is interchangeable with the statement that f is k-SOS

mod I whenever deg(f) ≤ d. These ideals, particularly the (1, k)-SOS ideals, are of

enormous interest in many application contexts. The study of (d, k)-SOS ideals was

first introduced by Lovasz in his 2003 paper [20]. In this paper Lovasz calls ideals with

this cooperative SOS structure “perfect” ideals and he asks, “Which ideals in R[x]

have the (1, 1)-SOS property?” The first progress toward an answer was contributed

by R. Thomas, P. Parrilo and J. Gouveia in [10]. We will discuss these results in

detail in the next chapter.

2.2 Polynomial Optimization

We illustrate a couple of standard techniques from polynomial optimization that will

be of use to us. Consider the optimization problem

minimize f(x)

such that gi(x) ≥ 0 for i = 1, . . . ,m(2.1)

First, we note that it is not a loss of generality to assume that the feasible region

is given by polynomial equations, as opposed to polynomial inequalities. For each gi

we define

g′i(x, si) = gi(x)− s2i

Then, we may rewrite the problem as

minimize f(x)

such that

g′i(x, si) = 0 for i = 1, . . . ,m

If we restrict our focus to real points (x1, . . . , xn, s1, . . . , sm) ∈ Rn+m then we may be

sure that the formulation above is equivalent to (2.1).

Secondly, we note that it is not a loss of generality to assume that the objec-

8


tive function for an optimization problem is linear. Again consider the optimization

problem given by

minimize f(x)

such that gi(x) ≥ 0 for i = 1, . . . ,m(2.2)

By introducing a variable z and including z − f(x) = 0 in the constraints we

obtain a polynomial optimization problem

minimize z

such that

z − f(x) = 0

gi(x) ≥ 0 for i = 1, . . . ,m

Note that this problem is equivalent to (2.2) and has a linear objective function.

This technique is very useful in the context of optimization via SOS formulations.

In particular, this allows us to rearrange a polynomial optimization problem into a

question about non-negativity of a linear polynomial. As such, the (1, 1)-SOS ideals

are of primary interest. We illustrate the utility of the (1, 1)-SOS property with an

example.

Example 2.2.1. Consider the elementary optimization problem

minimize y − x2

subject to

x, y ∈ {±1}

Of course, the answer is −2. We illustrate how SOS techniques can lead us to this

solution.

We rewrite the problem as follows:

maximize d

9


subject to

z − d ≥ 0

z − (y − x2) = 0

x2 − 1 = 0

y2 − 1 = 0

We’re interested in the values of d for which z − d is non-negative over the

variety of the ideal

I = 〈x2 − 1, y2 − 1, z − (y − x2)〉

We’ll construct a 4× 4 symmetric matrix Q = (qij) and a vector

u =

1

x

y

z

Let p denote the polynomial

p =

1

x

y

z

T

Q

1

x

y

z

Using Maple, we find the normal form of p in the quotient ring R[x, y, z]/I.

pnf = (2q23 + 2q24)xz + (2q12 + 2q23)x+

(2q13 + 2q14 − 2q34 − 2q44)z + q11 + 2q13 + q22 + q33

and we enforce the congruence

z − d ≡ pnf

10


This gives us the constraints

(2q23 + 2q24) = 0

(2q12 + 2q23) = 0

(2q13 + 2q14 − 2q34 − 2q44) = 1

Note that the constant term q11 + 2q13 + q22 + q33 plays the role of −d. Ultimately,

we have constructed the following equivalent problem:

Find a 4× 4 PSD matrix Q = (qij) that satisfies

(2q23 + 2q24) = 0

(2q12 + 2q23) = 0

(2q13 + 2q14 − 2q34 − 2q44) = 1

with minimal value of

q11 + 2q13 + q22 + q33

This is a semidefinite programming problem. We can solve this using, for example,

the SeDuMi package in MatLab. We obtain the result

Q =

1.1388 0 −0.1388 0.6388

0 0 0 0

−0.1388 0 1.1388 −0.6388

0.6388 0 −0.6388 0.6388

One can verify that this satisfies the constraints above. We obtain

min z = max d = −(q11 + 2q13 + q22 + q33) = −2

Notice that in this problem it is not necessary to explicitly construct the SOS rep-

resentation for our objective function. It is sufficient to know that the representation

11


exists. This is commonly the case in our applications. However, should the specific

SOS representation be desired it may be obtained from the Cholesky decomposition

of the matrix Q. The process above has been implemented in a MatLab environment

with the SOStools package [26].

In the example above we were able to use SOS techniques to find our solution

using the vector

u =

1

x

y

z

This vector worked because the ideal of constraints in the example was a (1, 1)-

SOS ideal. In general, however, it is entirely possible that the SOS representation of

a given polynomial requires several large monomials, and therefore a large vector u.

Consider the vector

u =

1

x

y

z

x2

y2

z2

xy...

x8y12z19

The SDP algorithms run in polynomial time on the size of the matrix Q. For the

u vector above the Q matrix is substantially larger and therefore the SDP solver will

run considerably slower. This can be a problem when studying whether or not SDP

formulations of polynomial optimization problems are competitive. We are motivated

to ask, “Is there a way to know when we are dealing with a (1, k)-SOS ideal?”

As demonstrated in the example, the best case scenario for SOS techniques in

optimization arises when the constraints of the problem form a (1, 1)-SOS ideal. In

this case, the Q matrix is optimally small and the SDP solvers run quickly and com-

12


petitively. Lovasz’s question of, “How can we recognize a (1, 1)-SOS ideal?” is the

primary focus of our research. In the next chapter we discuss the existing results due,

primarily, to R. Thomas, P. Parrilo, and J. Gouveia.

2.3 Real Radical Ideals

Before we proceed to the discussion of existing results we take a look at one more

object that will be of use to us. The real radical of an ideal is defined as follows:

Definition 2.3.1. Given an ideal I ⊆ R[x], the real radical of I is given by

R√I = {f(x) ∈ R[x] | f 2m(x) + σ(x) ∈ I, for some m ∈ N and σ ∈ Σ[x]}

We say an ideal I is real radical if I = R√I.

This definition may not be immediately easy to visualize. However, a real version

of Hilbert’s Nullstellensatz will provide some insight into the geometry of the real

radical. We refer to [2] for proof.

Theorem 2.3.2 (Real Nullstellensatz). Let I ⊆ R[x] be an ideal and let VR(I)

denote the real variety of I. Then the real radical of I is the vanishing ideal of

VR(I), i.e.R√I = {f ∈ R[x] | f(x) = 0 for all x ∈ VR(I)}

So, the real radical of an ideal I provides a way to eliminate as many complex

solutions as possible from the variety V(I). As a word of warning, we emphasize that

this removes as many complex points as possible from the variety. There are plenty

of situations in which all of the polynomials in a real radical ideal have complex roots

in common. In other words, even for a real radical ideal I we cannot be sure that

V(I) = VR(I). We provide an illustration below.

13


Example 2.3.3. Consider the unit circle in R2 and let I be the ideal 〈x2+y2−1〉 ⊂R[x, y]. It can be shown that I is real radical. However, the point (i,

√2) is one

of infinitely many complex points in V(I).

If we restrict our focus to zero dimensional ideals we may avoid this complication.

This is formalized in the following proposition. We first observe a simple lemma.

Lemma 2.3.4. If I is zero dimensional then R√I is zero dimensional.

Proof. This follows immediately from the observation that V(R√I) ⊆ V(I). �

We now state the proposition

Proposition 2.3.5. Let I ⊆ R[x] be a zero dimensional ideal. Then VR(R√I) =

V(R√I).

Proof. Let I ⊆ R[x] be a zero dimensional ideal. Let V(I) consist of the following

finite set of points

V(I) = {p1, . . . , pm} ⊆ Cn

By the preceding lemma, and without loss of generality, suppose that

VR(I) = {p1, . . . , pk} ⊆ Rn for k ≤ m

For each pi ∈ VR(I) we have a corresponding maximal ideal Mi ⊂ R[x] with

V(Mi) = {pi} for all i = 1, . . . , k. Recall that the union of varieties corresponds to

the intersection of ideals. Therefore, we have

V(M1 ∩ · · · ∩Mk) = {p1, . . . , pk} ⊆ Rn

By the Real Nullstellensatz (Theorem 2.3.2) we conclude that

R√I = M1 ∩ · · · ∩Mk

14


and the result follows. �

The art of properly formulating a problem is a nuanced and vital aspect of ef-

ficient optimization. At the surface, any collection of constraints that describe the

same feasible region is valid and the researcher must choose a presentation that is

conducive to her algorithms. If we regard the constraint polynomials as generators

of an ideal, the notion of valid formulations is equivalent to the notion of basis sets

of the ideal. Any collection of generators is permissible provided that the variety is

unchanged. The strength of the real radical of an ideal lies in the ability to shrink

the variety, and thus often simplify the problem, without altering the real variety. A

complex solution is impractical and undesirable in the majority of optimization con-

texts. Hence, when we remodel a formulation of an optimization problem we strive

to leave the feasible region unchanged, but more precisely we strive to leave the real

feasible region unchanged. This is exactly what the real radical allows us to do. As

a simple example consider the optimization problem

minimize f(x, y, z)

such that

x2 + y4 + z6 = 0

If we are working over R3 then this, of course, is equivalent to the much simpler

problem

minimize f(x, y, z)

such that

x = y = z = 0

In algebraic terms, what we have done is replace the ideal of constraints from the

first formulation with its real radical. This technique is also of use in the study of

feasibility. Consider the problem

minimize f(x, y, z)

such that

xy2 + yz + 1 = 0

15


x2 + y2 + xz − 2 = 0

x4 + y2z2 + z4 = 0

If we gather the constraints into an ideal I and compute the real radical we see thatR√I = 〈1〉. Therefore, although complex solutions exist, the problem is infeasible

over R3.

There are algorithms for computing real radical ideals that have been implemented

in some computer algebra software packages. As real radical ideals will be so vital to

our study we take a moment to discuss the specifics of one of these algorithms. The

following is a summary of a technique proposed by J. Lasserre and M. Laurent in [16].

Consider a sequence indexed by n-tuples of non-negative integers

(yα)α∈Nn ∈ RNn

We define the moment matrix M(y) as follows

M(y) = (yα+β)α,β∈Nn

The moment matrices that are positive semidefinite will be of interest to us. We

will use the notation M(y) � 0 to indicate that that M(y) is PSD. Consider xα =

xα11 x

α22 · · ·xαn

n . It can be helpful to think of the sequence y and the associated moment

matrix M(y) as being indexed by the monomials in R[x].

Given a polynomial h ∈ R[x] let hα denote the coefficient of xα in h. We then

associate h with the infinite column vector

vec(h) = (hα)α∈Nn

By abuse of language we will say that h lies in the kernel of M(y) when vec(h) does.

Also, given a polynomial h we construct a new sequence hy as follows:

hy = M(y)vec(h) ∈ RNn

16


We now present one of Lasserre’s and Laurent’s principal results. We refer to [16] for

proof.

Theorem 2.3.6. Let I = 〈h1, . . . , hm〉 be an ideal and assume that VR(I) is finite.

If

M(y) � 0, and M(hiy) = 0 for i = 1, . . . ,m

then the kernel of M(y) is a real radical ideal, rankM(y) ≤ |VR(I)| and R√I ⊆

KerM(y), with equality if and only if M(y) has maximum rank equal to |VR(I)|.

In order to develop an algorithm based on the preceding result we need to introduce

a truncation of the infinite moment matrices. Given an integer t, let Mt(y) denote

the principal submatrix of M(y) indexed by the set

Tn,t = {α ∈ Nn |n∑i

αi ≤ t}

We also define the following for an ideal I = 〈h1, . . . , hm〉

dj = ddeg(hj)/2e, d = maxj=1,...,m

dj

The following theorem guides the SDP algorithm. See [16] for proof.

Theorem 2.3.7. Fix t ≥ d and assume Mt(y) is a maximum rank matrix satisfying

Mt(y) � 0, and Mt−di(hiy) = 0 for i = 1, . . . ,m

If we also have

rankMt(y) = rankMt−d(y)

then R√I = KerMt(y).

We close this discussion with an example.

17


Example 2.3.8. Let I ⊆ R[x1, x2] be generated by h1 and h2 where

h1(x1, x2) = x1(x21 + 1) and h2(x1, x2) = x2(x

22 + 1)

It is not difficult to see that the origin (0, 0) is the only real root for the polyno-

mials above. We will show how Lasserre’s and Laurent’s technique can lead us to

this conclusion. Notice that |V(I)| = 9. Also note that d1 = d2 = 2. We’ll take

t = 3.

We consider a sequence (yα)α∈N2 and the associated truncated moment matrix

M3(y). We will organize our work according to graded lex order with x1 > x2

and we will include the indices to aid understanding. We obtain the following for

M3(y)

(0,0) (0,1) (1,0) (0,2) (1,1) (2,0) (0,3) (1,2) (2,1) (3,0)

(0,0) y(0,0) y(0,1) y(1,0) y(0,2) y(1,1) y(2,0) y(0,3) y(1,2) y(2,1) y(3,0)

(0,1) y(0,1) y(0,2) y(1,1) y(0,3) y(1,2) y(2,1) y(0,4) y(1,3) y(2,2) y(3,1)

(1,0) y(1,0) y(1,1) y(2,0) y(1,2) y(2,1) y(3,0) y(1,3) y(2,2) y(3,1) y(4,0)

(0,2) y(0,2) y(0,3) y(1,2) y(0,4) y(1,3) y(2,2) y(0,5) y(1,4) y(2,3) y(3,2)

(1,1) y(1,1) y(1,2) y(2,1) y(1,3) y(2,2) y(3,1) y(1,4) y(2,3) y(3,2) y(4,1)

(2,0) y(2,0) y(2,1) y(3,0) y(2,2) y(3,1) y(4,0) y(2,3) y(3,2) y(4,1) y(5,0)

(0,3) y(0,3) y(0,4) y(1,3) y(0,5) y(1,4) y(2,3) y(0,6) y(1,5) y(2,4) y(3,3)

(1,2) y(1,2) y(1,3) y(2,2) y(1,4) y(2,3) y(3,2) y(1,5) y(2,4) y(3,3) y(4,2)

(2,1) y(2,1) y(2,2) y(3,1) y(2,3) y(3,2) y(4,1) y(2,4) y(3,3) y(4,2) y(5,1)

(3,0) y(3,0) y(3,1) y(4,0) y(3,2) y(4,1) y(5,0) y(3,3) y(4,2) y(5,1) y(6,0)

We next compute the vectors corresponding to h1(x1, x2) = x1(x21 + 1) and

18


h2(x1, x2) = x2(x22 + 1). Again, for graded lex order with x1 > x2, we have

vec(h1) =

0

0

1

0

0

0

0

0

0

1

0...

and vec(h2) =

0

1

0

0

0

0

1

0

0

0

0...

Notice that the product h1y = M(y)vec(h1) is a sequence in which the ith element

is the sum of the third and tenth entries of the ith row of M(y).

h1y = {(y(1,0) + y(3,0)), (y(1,1) + y(3,1)), (y(2,0) + y(4,0)), (y(1,2) + y(3,2)),

(y(2,1) + y(4,1)), (y(3,0) + y(5,0)), . . . }

Similarly, for h2y we have

h2y = {(y(0,1) + y(0,3)), (y(0,2) + y(0,4)), (y(1,1) + y(1,3)), (y(0,3) + y(0,5)),

(y(1,2) + y(1,4)), (y(2,1) + y(2,3)), . . . }

Next we construct the moment matrices for the hiy sequences. Recall that

we are interested in the truncation Mt−di(hiy). For this example, t = 3 and

d1 = d2 = 2 so we compute the first truncation. We have

M1(h1y) =

(y(1,0) + y(3,0)) (y(1,1) + y(3,1)) (y(2,0) + y(4,0))

(y(1,1) + y(3,1)) (y(1,2) + y(3,2)) (y(2,1) + y(4,1))

(y(2,0) + y(4,0)) (y(2,1) + y(4,1)) (y(3,0) + y(5,0))

19


and

M1(h2y) =

(y(0,1) + y(0,3)) (y(0,2) + y(0,4)) (y(1,1) + y(1,3))

(y(0,2) + y(0,4)) (y(0,3) + y(0,5)) (y(1,2) + y(1,4))

(y(1,1) + y(1,3)) (y(1,2) + y(1,4)) (y(2,1) + y(2,3))

In accordance with Theorem 2.3.7 we now enforce

M3(y) � 0 and M1(hiy) = 0 for i = 1, 2

Since M1(h1y) = 0 we may claim that y(2,0) = −y(4,0). Likewise, M1(h2y) = 0 tells

us that y(0,2) = −y(0,4). Note that the elements of y above are diagonal elements

of M3(y). Since we’re assuming M3(y) is positive semidefinite the diagonal entries

must be non-negative. Hence,

y(2,0) = y(4,0) = y(0,2) = y(0,4) = 0

This, and the fact that M3(y) is positive semidefinite, produces the following claim:

yα = 0 for all α 6= (0, 0)

Therefore, the maximum rank of M3(y) is 1. We point out that rankM3(y) =

|VR(I)| as expected. In terms of standard basis vectors ei we have

KerM3(y) = span{ei : i = 2, . . . , 10}

Discarding redundant generators we obtain

R√I = 〈x1, x2〉

As anticipated, this is the real radical ideal corresponding to the origin in R2.

Singular, the computer algebra software, has an optional “realrad.lib” package

that allows one to compute the real radical of a given ideal. For a thorough guide to

the use of Singular see [11].

20


We warn that the running time for the real radical algorithms can be very difficult

to predict. The real radical approach to the simplification of optimization problems

works beautifully at times, but in many other cases the algorithm to compute the

real radical can be prohibitively expensive. The nuances of these cases are still being

explored [17]. Real radical ideals also play an important role in the SOS approach to

optimization. We discuss this idea in the next section.

21


Chapter 3

Existing Classification

3.1 Theta Bodies

The concept of real radical ideals will allow us to illuminate the connection between

the algebraic notion of (1, k)-SOS ideals and the geometric idea of the theta body

sequence for an ideal. This is our next definition.

Definition 3.1.1. Given an ideal I ⊆ R[x] and a positive integer k we define the

kth theta body for I, denoted Θk(I), as follows

Θk(I) = {x ∈ Rn | `(x) ≥ 0 for all linear polynomials ` that are k-SOS mod I}

Let a and b be positive integers with a < b. Notice that Σa[x] ⊆ Σb[x]. Consequently,

we have Θa(I) ⊇ Θb(I) for all a, b ∈ N with a < b. More importantly, for any positive

integer k we have Θk(I) ⊇ VR(I). This gives us the sequence

Θ1(I) ⊇ Θ2(I) ⊇ · · · ⊇ VR(I)

In order to study this sequence we make the following observation regarding the

closure of the convex hull of the variety of an ideal.

cl(conv(VR(I))) = {x ∈ Rn | `(x) ≥ 0 for all linear ` that are non-negative over VR(I)}

In other words, the closure of the convex hull of VR(I) may be regarded as the

intersection of all half-spaces containing VR(I). Notice that being k-SOS mod Iis sufficient for being non-negative over VR(I). As a result, we may improve our

22


sequence above as follows

Θ1(I) ⊇ Θ2(I) ⊇ · · · ⊇ cl(conv(VR(I)))

The convergence (or lack thereof) of this sequence may be regarded as the ge-

ometric counterpart of the algebraic notion of (1, k)-SOS ideals. We formalize this

notion with the following theorem from [10].

Theorem 3.1.2. Given an ideal I ⊆ R[x], if I is (1, k)-SOS for some positive

integer k then Θk(I) = cl(conv(VR(I))).

Proof. Suppose that I is (1, k)-SOS. Since cl(conv(VR(I))) is a closed, convex set

we know that given a point p /∈ cl(conv(VR(I))) we may construct a hyperplane

`(x) that separates p from cl(conv(VR(I))). In particular, we may choose ` such

that `(p) < 0 and `(x) ≥ 0 for all x ∈ cl(conv(VR(I))). So ` is k-SOS mod I. Thus,

since `(p) < 0, we know that p /∈ Θk(I). Therefore, Θk(I) ⊆ cl(conv(VR(I))). The

reverse containment always holds. �

A great deal of the work in [10] serves to establish that the converse of Theo-

rem 3.1.2 holds when I is a real radical ideal. The paper also establishes a geometric

characterization of (1, 1)-SOS ideals. We review some concepts from polyhedral op-

timization that will assist our exploration of these results.

3.2 Polyhedra in Rn

The majority of our work will focus on polytopes. These are the bounded polyhedra

in Rn.

Definition 3.2.1. A polytope is the convex hull of a finite set of points in Rn. We

will call a polytope P binary if all of the vertices of P are points in {±1}n.

23


Notice that we will use the term binary to refer to {±1} points instead of the

more traditional {0, 1} points. We do this to facilitate some isometry shortcuts that

we will introduce in the next section.

Optimization over polytopes has a rich and interesting history with an extensively

researched body of work. We will utilize several of these results in our exploration of

SOS ideals. The reader is referred to [28] for a tour of the principles of polyhedral

optimization. We introduce some terminology that will frequently arise in our study.

Definition 3.2.2. Let P ⊆ Rn be a polytope. A linear inequality `(x) ≥ 0 is

called a valid inequality for P if `(p) ≥ 0 for all p ∈ P .

The notion of valid inequalities will allow us to formalize many claims about the

geometric properties of polytopes and varieties. We can use valid inequalities to

introduce two exceptionally useful concepts. These are presented below.

Definition 3.2.3. Let P ⊆ Rn be a polytope and let `(x) ≥ 0 be a valid inequality

for P . The set F given by

F = {p ∈ P | `(p) = 0}

is called a face of P . We say that ` is a face-defining inequality corresponding to

F .

We may occasionally abuse language and refer to ` itself as a face of P . Be aware

that for P ⊆ R3 the faces of P are not necessarily the 2-dimensional faces. For

example, the vertices of P are considered to be faces of P . Likewise, the empty set ∅is a face of every polytope, as is P itself (by virtue of the trivial inequality 0x ≥ 0).

The intuitive notion of 2-dimensional faces of 3-dimensional polytopes is made formal

in the following definition.

Definition 3.2.4. Let P ⊆ Rn be a polytope with dim(P ) = k. Let F be a face

of P . We say that F is a facet of P if dim(F ) = k − 1. If F is given by a valid

inequality `(x) ≥ 0 we call ` a facet-defining inequality.

24


3.3 Geometry of (1, 1)-SOS Ideals

We are now prepared to study the existing classification of (1, 1)-SOS ideals. The

following theorem from [10] provides a geometric characterization of the polytopes

that are given by the convex hull of the varieties of zero-dimensional (1, 1)-SOS ideals.

Theorem 3.3.1. Let P ⊆ Rn be a polytope and let vert(P ) be the set of vertices

of P . Let I be the real radical ideal with VR(I) = vert(P ). Then I is (1, 1)-SOS if

and only if P can be expressed in terms of facet-defining inequalities ì(x) ≥ 0 for

i = 1, . . .m where `2i (x) ≡ ì(x) mod I.

Proof. Let the facets of P be given by linear polynomials `1(x), . . . , `k(x) where

`2i (x) ≡ ì(x) for all i. One of the many variations of Farkas’ Lemma [23] tells us

that any valid inequality for P can be expressed as a non-negative linear combina-

tion of the facet-defining inequalities for P . Hence, given a linear polynomial f(x)

with f(x) ≥ 0 over P we can always write

f(x) = α1`1(x) + · · ·+ αk`k(x) ≡ α1`21(x) + · · ·+ αk`

2k(x) mod I

for non-negative αi ∈ R. Hence, f(x) is 1-SOS mod I meaning I is (1, 1)-SOS.

Conversely, suppose that I is (1, 1)-SOS. Let `(x) ≥ 0 define a facet F of P . Since

`(x) is non-negative over P we know that there exist linear polynomials fi(x) such

that `(x) ≡ f 21 (x) + · · ·+ f 2

k (x) mod I. However, since f 2i (x) is non-negative over

Rn we know that since `(p) = 0 for all p ∈ F we must have fi(p) = 0 for each

i = 1, . . . , k. In particular, fi(p) = 0 for each i over the vertices of F . Since the

vertices of F comprise sufficiently many points to define the hyperplane `(x) = 0

we see that each fi(x) must define this hyperplane as well. Hence, for some α ∈ Rwe must have `(x) ≡ α`2(x) mod I. Therefore, the linear polynomial `′(x) = 1

α`(x)

defines F and is idempotent in R[x]/I. �

We provide an example that illustrates how one might use this theorem.

Example 3.3.2. Consider the binary polytope P ⊆ R3 pictured below.

25


(1,−1,−1)

(−1,−1, 1)

(1, 1, 1)

(1, 1,−1)

Figure 3.1: The polytope for a (1, 1)-SOS ideal

We compute the following ideal

I = 〈x2 − 1, y2 − 1, z2 − 1, xz − yz + x− y, xy + x− y − 1〉

Notice that I is a real radical ideal and VR(I) = vert(P ). We next compute the

facets of the polytope P . We obtain

−x+ 1 ≥ 0, y + 1 ≥ 0, z + 1 ≥ 0

x− y ≥ 0, −x+ y − z + 1 ≥ 0

We now test for idempotence of these facets in the quotient ring R[x]/I. For

example, we see that

(−x+ y − z + 1)2 ≡ −2x+ 2y − 2z + 2 mod I

Note that if `(x) ≥ 0 is a facet-defining inequality for a facet F then for any

positive scalar α ∈ R+ the inequality α`(x) ≥ 0 defines F as well. In this example

we scale the inequality above by 12

(note that it might be necessary to reverse the

inequality if α < 0). This yields(1

2

)(−x+ y − z + 1) ≥ 0

26


It can be shown that this linear polynomial is idempotent in R[x]/I. We may

proceed to show that, after appropriate scaling, all of the facet-defining half-spaces

above are idempotent mod I. Therefore, I is (1, 1)-SOS.

We provide an additional example of an ideal that is not (1, 1)-SOS.

Example 3.3.3. Consider the polytope P ⊆ R3 pictured below.

(1,−1,−1)

(−1,−1, 1)

(1, 1, 1)

(1, 1,−1)

(−1, 1,−1)

Figure 3.2: The polytope of an ideal that is not (1, 1)-SOS

We first compute the facets of P

−x+ 1 ≥ 0, −y + 1 ≥ 0, z + 1 ≥ 0

−x+ y − z + 1 ≥ 0, x− y − z + 1 ≥ 0, x+ y + z + 1 ≥ 0

Next we compute the real radical ideal I corresponding to the polytope P . We

have

I = 〈x2 − 1, y2 − 1, z2 − 1, xz − yz + x− y, xy + yz − x− z〉

We now test for idempotence of the facet-defining linear polynomials. Let us

consider the facet F given by −x + y − z + 1 ≥ 0. Computing the normal form

(here we use graded lexicographic order with x > y > z) we note that

(−x+ y − z + 1)2 ≡ 2yz − 6x+ 4y − 4z + 4 mod I

27


So the normal form of (−x+ y− z + 1)2 with respect to a graded monomial order

is quadratic. This is not a problem we can correct with scaling. We conclude

that there is no facet-defining inequality for F that grants us an idempotent linear

polynomial in R[x]/I. Therefore, I cannot be (1, 1)-SOS.

As demonstrated above, this theorem provides a reliable way to test whether or

not a zero-dimensional ideal has the (1, 1)-SOS property. However, the process can

be very labor intensive. If we are given a polytope - be it a list of vertices, a half-

space representation Ax ≤ b, or simply a drawing - we are required to compute the

corresponding ideal in order to test the facets for idempotence. However, given an

arbitrary polytope, it is not in general obvious what the facet-defining inequalities

are. Moreover, the corresponding ideal is not necessarily easy to construct. In the

opposite direction, suppose we are handed an ideal and asked if it is (1, 1)-SOS. We

must compute the real variety of the ideal, often an expensive task, and we must

find the facets of the polytope given by the variety’s convex hull. Again, this can be

difficult for general cases. Once the facets are collected we compute a Grobner basis

and study the reduction of the facet-defining linear polynomials. No stage of this

process is trivial, in general, and we emphasize that a Grobner basis for the ideal is

always necessary regardless of how the problem is originally presented.

The primary focus of our work is to develop a series theorems that will allow us

to recognize the (1, 1)-SOS property of an ideal I simply by inspecting the generators

of a Grobner basis of I. The ability to efficiently and reliably verify such a helpful

property without the need for facets, real varieties, or other geometric computations

will be of great value to researchers in optimization and pure mathematics. Our

results are presented in the following chapters.

28


Chapter 4

Preliminary Observations and Technical Lemmas

In this section we will make a series of observations that will justify some claims and

shortcuts that we will employ in our analysis. One of the most useful classification

tools is the invariance of SOS structure among isometric polytopes. We will begin this

section with a discussion of this concept. First we make some elementary definitions.

Definition 4.0.4. Let I ⊆ R[x] = R[x1, . . . , xn] be an ideal. We say that I is a

binary ideal if x2i − 1 lies in I for all i = 1, . . . , n.

As with the analogous definition for polytopes, we emphasize that we will use the

term binary to refer to the {±1} case instead of the more traditional {0, 1} binary

notion. The reason for this will be made clear momentarily. We also make the

following intuitive definition.

Definition 4.0.5. Let I ⊆ R[x] be a zero-dimensional real radical ideal. We call

P = conv(VR(I)) the polytope corresponding to I. Similarly, given a polytope

P ⊆ Rn we call the vanishing ideal of vert(P ) the ideal corresponding to P . Notice

that this ideal is necessarily zero-dimensional and, by the Real Nullstellensatz

(Theorem 2.3.2) it is real radical.

4.1 Isometries of the Cube

When P is a polytope corresponding to an ideal I we may occasionally abuse lan-

guage and say that P is a (1, 1)-SOS polytope when I is a (1, 1)-SOS ideal. We are

now prepared to discuss the shortcuts granted by isometric polyhedra. Consider the

polytopes pictured below

29


(1,−1,−1)

(−1,−1, 1)

(1, 1, 1)

(1, 1,−1)

(1,−1,−1)

(1, 1,−1)

(1,−1, 1)

(−1, 1, 1)

(−1, 1,−1)

Figure 4.1: Isometric polytopes

These are isometric polytopes. We aim to establish that a polytope is (1, 1)-SOS

if and only if all isometric polytopes are (1, 1)-SOS. Our proof will be assembled from

a series of small lemmas that we presently introduce. First, we quickly review some

concepts about isometries of polygons in Rn. In general, an isometry is a map between

metric spaces that preserves distance between points. When we regard a point x ∈ Rn

as a vector, we may regard an isometry of Rn as a map x 7→ Ax+s where A is a n×nmatrix satisfying ATA = I. Since A and AT have the same determinant it follows

that det(A) = ±1. If det(A) = 1 we call the isometry a rotation. If det(A) = −1 we

call the isometry a reflection. Consider the cube in R3 with vertices in {±1}3. We

use this cube rather than the traditional [0, 1]3 unit cube because the [−1, 1]3 cube

is centered at the origin an hence is more cooperative with the study of linear maps.

Regarding this cube in R3 we have the following 48 isometries that map the cube to

itself: ±1 0 0

0 ±1 0

0 0 ±1

,

±1 0 0

0 0 ±1

0 ±1 0

,

0 ±1 0

±1 0 0

0 0 ±1

0 ±1 0

0 0 ±1

±1 0 0

,

0 0 ±1

±1 0 0

0 ±1 0

,

0 0 ±1

0 ±1 0

±1 0 0

As such, our isometries are signed permutations in R3. We will use the concept of

30


isometries and signed permutations in both an algebraic and geometric context. For

example, we will regard a transformation on R3 given by

(x0, y0, z0) 7→ (y0,−z0, x0)

as well as a map R[x, y, z]→ R[x, y, z] given by

f(x, y, z) 7→ f(y,−z, x)

interchangeably as isometries and signed permutations. We will call two polytopes

P1 and P2 isometric if there exists an isometry matrix A that surjectively maps

A : vert(P1) 7→ vert(P2)

We may occasionally refer to a pair of zero-dimensional real radical ideals as

isometric when their corresponding polytopes are isometric, i.e. when they differ by

a signed permutation of variables. We now point out an obvious yet useful notion.

The proof is immediate and will be suppressed.

Lemma 4.1.1. Let xα and xβ be monomials in R[x]. If there exists a signed

permutation σ such that σ(xα) = xβ then |α| = |β|.

In other words, a signed permutation cannot change the degree of a monomial,

and therefore cannot change the total degree of a polynomial. In order to establish

our claim we will need to explore some principles of idempotent linear polynomials

in a quotient ring. We begin with a simple observation about vertices of polytopes.

Lemma 4.1.2. Let P ⊆ Rn be a binary polytope. If a binary point b ∈ {±1}n lies

in P then b is a vertex of P .

Proof. Recall that a point b is a vertex of a polytope P if and only if b cannot be

written as a non-trivial convex combination of two other points of P . Indeed, this

is often taken as the definition of a vertex. As P is the convex hull of its vertices,

it is sufficient to restrict our focus to the vertices of P . That is, we will show that

31


a binary point b ∈ P cannot be written as a non-trivial convex combination of two

points p, q ∈ vert(P ). Suppose

b = λ1p+ λ2q

where λ1 and λ2 are non-negative real numbers with λ1 + λ2 = 1. Notice that the

result is immediate if p = q. Suppose that p 6= q. Then for some coordinate i we

know pi 6= qi. Since b is binary we have

λ1pi + λ2qi = ±1

This means that λ1−λ2 = ±1. However, we also know that λ1+λ2 = 1. Therefore,

either λ1 = 0 or λ2 = 0. In either case, the convex combination is trivial. Hence,

b is a vertex of P . �

4.2 Idempotence of Affine Polynomials

We first justify a claim regarding idempotence of linear polynomials that we will use

very frequently.

Theorem 4.2.1. Let P ⊆ Rn be a binary polytope and let I ⊆ R[x] be the corre-

sponding ideal. Let f(x) ≥ 0 define a facet F of P and let `(x) ∈ R[x] be a linear

polynomial. If f 2 ≡ ` mod I then there exists a linear polynomial f ′(x) defining

F such that f ′ is idempotent mod I.

Proof. Since f(v) = 0 for all vertices v of F we know by the preceding discussion

that `(v) = 0 for all vertices of F . Since ` is linear and agrees with f on the

vertices of F , it follows that ` and f define the same hyperplane. Hence, ` is a

scalar multiple of f . Let ` = 1αf . Define f ′ = αf and notice that f ′(x) also defines

F . Then

(f ′)2 = α2f 2 ≡ α2` = αf = f ′ mod I

Therefore, f ′ is a linear polynomial that defines the facet F and f ′ is idempotent

in R[x]/I. �

32


This theorem allows us to somewhat relax the procedure to test for the (1, 1)-SOS

property outlined by Theorem 3.3.1. We offer the following corollary.

Corollary 4.2.2. Let P ⊆ Rn be a binary polytope with corresponding ideal I ⊆R[x]. Then I is (1, 1)-SOS if and only if any representation of P by facets gi(x) ≥0 satisfies g2i ≡ ì mod I where ì ∈ R[x] is (affine) linear for all i.

Proof. This follows immediately from Theorem 4.2.1. �

So we have established that the (1, 1)-SOS property of an ideal I hinges upon the

ability to reduce certain quadratic polynomials to the linear level mod I. We will

dissect this notion in detail in the following chapter. For now, we give a name to this

near-idempotent behavior.

Definition 4.2.3. Let I ⊆ R[x] be an ideal and let f ∈ R[x] be a polynomial. We

will call f pseudo-idempotent if f 2 ≡ αf mod I for some α ∈ R.

We are now ready to establish our claim regarding isometric polytopes. Some

of our claims about the SOS structure of low-dimensional polytopes can be justified

with exhaustive computations. The following proposition will significantly reduce the

number of polytopes we must consider in these cases.

Proposition 4.2.4. Let P1 and P2 be isometric polytopes with corresponding ideals

I1 and I2 respectively. Then I1 is (1, 1)-SOS if and only if I2 is (1, 1)-SOS.

Proof. We have established that the isometry that relates P1 and P2 may be re-

garded algebraically as a signed permutation of variables. By Theorem 4.2.1 we

know that all of the facet-defining linear polynomials for P1 are pseudo-idempotent

mod I1. By Lemma 4.1.1, it follows that the facet-defining polynomials for P2 are

pseudo-idempotent mod I2 as well. The result then follows from Corollary 4.2.2.

�

33


4.3 Constructing the Ideal of a Binary Polytope

We now discuss a simple tool that will allow us to construct the ideal corresponding

to a given polytope. We define exclusion polynomials as follows.

Definition 4.3.1. Let p = (p1, . . . , pn) be a binary point in {±1}n. We define the

exclusion polynomial, denoted φp(x) ∈ R[x], corresponding to p as follows:

φp(x) = (x1 + p1)(x2 + p2) · · · (xn + pn)

The exclusion polynomials will provide a way to construct the ideal correspond-

ing to a polytope P by studying which binary points are not in P . The following

proposition clarifies this notion.

Proposition 4.3.2. Let I ⊆ R[x] be a radical, binary ideal. Let p be a binary

point in Rn. Then φp ∈ I if and only if p /∈ V(I).

Proof. Suppose that φp ∈ I. Since by construction we have φp(p) 6= 0 we know

that p /∈ V(I). Conversely, suppose that p /∈ V(I). Let q = (q1, . . . , qn) be a point

in V(I) and notice that q must be binary since x2i − 1 ∈ I for all i = 1, . . . , n.

Since q 6= p we know that for some coordinate i we must have qi 6= pi. In other

words, qi = −pi. Therefore, φp(q) = 0. Since q was an arbitrary point in V(I) and

I is radical, we conclude that φp ∈ I. �

We offer a quick example.

Example 4.3.3. Suppose we are given the polytope P pictured below

Note that the binary points in R3 that are absent from P are

(1,−1, 1), (1, 1, 1), and (−1, 1, 1)

34


(1,−1,−1)

(−1,−1, 1)

(1, 1,−1)

(−1, 1,−1)

Figure 4.2: Using elimination polynomials

Consequently, we construct the binary ideal

I = 〈x21 − 1, x22 − 1, x23 − 1, (x1 + 1)(x2 − 1)(x3 + 1),

(x1 + 1)(x2 + 1)(x3 + 1), (x1 − 1)(x2 + 1)(x3 + 1)〉

This is the ideal corresponding to P .

Our claim at end of the previous example requires a bit of justification. Recall

that the ideal corresponding to a polytope must be zero-dimensional and real radical.

Certainly, any binary ideal contains a univariate polynomial in each variable by con-

struction. So all binary ideals are zero dimensional. The remainder of the argument

follows below.

Lemma 4.3.4. Let I be a binary ideal in R[x] given by

I = 〈x21 − 1, . . . , x2n − 1, φp1 , . . . , φpk〉

where φp1 , . . . , φpk are exclusion polynomials for binary points p1, . . . , pk in Rn.

Then I is a real radical ideal.

Proof. Notice that since I contains x2i − 1 for all i we know that all points of V(I)

35


are real. So, as a consequence of the Real Nullstellensatz (Theorem 2.3.2), we can

say that if I is radical then I is real radical. Recall that the traditional algorithm

to compute the radical of a zero dimensional ideal involves adjoining the squarefree

parts of the minimal univariate polynomials for each variable. See, for example, [7]

for details. In the case of our ideal I, the univariate polynomials are immediately

available, the x2i − 1 generators. As these polynomials are clearly squarefree, we

conclude that I is radical. Therefore, I is real radical. �

So we now have a method for computing the unique real radical ideal correspond-

ing to a given polytope. Notice that ideals constructed in accordance with this method

are not in general going to be in Grobner basis form. A Grobner basis will be necessary

for much of our analysis. Unless otherwise specified, all references to Grobner bases

herein will be with respect to graded lexicographic order with x1 > x2 > · · · > xn. A

graded order is enormously useful for our purposes but, apart from being graded, the

specifics of the monomial order chosen are otherwise unimportant.

The observations presented in this section will provide the preliminary groundwork

we’ll need to explore the algebraic classification of (1, 1)-SOS ideals. We will begin in

R3 where we have the luxury of visible graphics and the ability to verify claims with

exhaustive computations when needed.

36


Chapter 5

Algebraic Characterization of (1, 1)-SOS Ideals

Our efforts to algebraically classify the (1, 1)-SOS property will make regular use of

Corollary 4.2.2 from the previous section. This corollary establishes the idea that the

SOS structure of an ideal lies in the ability to reduce certain quadratic polynomials

to the linear level via division by the generators of a Grobner basis for the given

ideal. Moreover, since our binary ideals will contain the polynomial x2i − 1 for all

i = 1, . . . , n, we need not be concerned about the reduction of any αx2i monomials in

these quadratic polynomials. It is the bilinear terms - the terms of the form xixj -

that are critical in this study. Many of our endeavors in this section aim to illuminate

exactly which arrangements of bilinear terms in the generators of the ideal ensure (or

negate) the (1, 1)-SOS property. To be thorough we state the following definition.

Definition 5.0.5. Let αxixj ∈ R[x] be a monomial for non-zero α ∈ R. We say

that αxixj is a bilinear term if i 6= j.

5.1 Distinguished Facets and Generators

We will begin our discussion with some properties of the [−1, 1]n hypercube in Rn.

All other binary ideals are subsets of this cube and we can use some of the features

of the cube to assist in our subsequent analysis.

Lemma 5.1.1. Every hypercube of the form [−1, 1]n is (1, 1)-SOS.

Proof. Let P ⊆ Rn be the hypercube [1, 1]n. Then all facets of P have the form

±xi + 1 ≥ 0. Let I be the corresponding ideal given by I = 〈x21 − 1, . . . , x2n − 1〉.

37


Note that by simply subtracting x2i − 1 we obtain

(±xi + 1)2 = x2i ± 2xi + 1 ≡ ±2xi + 2 mod I

Hence, by Corollary 4.2.2 we conclude that P is (1, 1)-SOS. �

As an immediate corollary to the above argument we have the following. The

proof is identical to the proof of the above lemma.

Corollary 5.1.2. Let P ⊆ Rn be a binary polytope with corresponding ideal I ⊆R[x]. Let F be a facet of P given by ±xi + 1 ≥ 0. Then F is pseudo-idempotent

mod I.

In our efforts to characterize the (1, 1)-SOS property we will regularly need to

demonstrate pseudo-idempotence of the facets of a given polytope. The above corol-

lary allows us to discard the facets that are given by inequalities that also define

facets of the hypercube in Rn. This motivates the following definition.

Definition 5.1.3. Let f(x) ≥ 0 define a facet F of a full-dimensional binary

polytope P . We say that F is a compliant facet if f is always pseudo-idempotent

mod I where I is any ideal corresponding to a binary polytope that has a facet

defined by f .

So Corollary 5.1.2 states that facets given by ±xi + 1 ≥ 0 are compliant. Later

we will see examples of other compliant facets. Since the facets that lie on the

hyperplanes ±xi + 1 = 0 will always be pseudo-idempotent mod any binary ideal we

will frequently ignore them in our study. This leads us to the following definition.

Definition 5.1.4. Let P ⊆ Rn be a binary polytope and let F be a facet of P .

We will say that F is a distinguished facet if F is not defined by ±xi + 1 ≥ 0 for

any i = 1, . . . , n.

38


The distinguished facets will be the interesting facets in our classification. Like-

wise, every binary ideal will contain the polynomials x2i −1 for all i. As our work will

almost exclusively use graded monomial orders and we will restrict our focus to full-

dimensional polytopes, these polynomials will always show up in the generating sets

for our ideals. As such, it will often be useful to isolate the “interesting” generators.

We have the following definition.

Definition 5.1.5. Let I ⊆ R[x] be a binary ideal in Grobner basis form for

a graded monomial order. Let g(x) be a generator of I. We say that g is a

distinguished generator of I if g 6= x2i − 1 for any i = 1, . . . , n.

We take a moment to discuss a handful of elementary observations. Note that

if I is the ideal corresponding to a polytope P then VR(I) = vert(P ). By the Real

Nullstellensatz (Theorem 2.3.2), it follows that the ideal corresponding to a given

polytope is unique, and vice versa. Consequently, we know that a polytope P ⊆ Rn

is the entire hypercube [−1, 1]n if and only if the corresponding ideal is given by

〈x21 − 1, . . . , x2n − 1〉. As this case is of little interest we offer the following definition.

Definition 5.1.6. Let P ⊆ Rn be a binary polytope. We will say that P is proper

if P is not the entire hypercube [−1, 1]n.

We point out that a proper binary polytope must have at least one distinguished

facet and the corresponding ideal must have at least one distinguished generator.

Also, notice that for an ideal corresponding to a full-dimensional polytope any dis-

tinguished generator must depend on at least two variables. The same is true of the

linear polynomials defining distinguished facets of full-dimensional polytopes. The

variables that appear in the facet-defining polynomials and ideal generators will be

of great use to us. We avoid trivialities with the following definition.

Definition 5.1.7. Let f(x) ∈ R[x] be a polynomial. Given a variable xi, we say

that f properly depends on xi if f has a monomial axα where a 6= 0 and αi 6= 0.

39


These observations will help us recognize the SOS structure of a given binary

ideal. We illustrate the basic idea of our approach with a simple example.

Example 5.1.8. Consider the polytope P pictured below.

(1,−1,−1)

(−1,−1, 1)

(1,−1, 1)

(1, 1,−1)

(−1, 1,−1)

(−1, 1, 1)

Figure 5.1: The polytope for an ideal with no quadratic distinguished generators

Following the procedure outlined in Example 4.3.3 we construct the exclusion

polynomial φp(x) for each of the binary points that is missing from vert(P ). In

this case, the only missing point is (1, 1, 1). We have the ideal

I = 〈x21 − 1, x22 − 1, x23 − 1, (x1 + 1)(x2 + 1)(x3 + 1)〉

It can be shown that this ideal is already in Grobner basis form for graded lexico-

graphic order with x1 > x2 > x3. Our only distinguished generator is

(x1 + 1)(x2 + 1)(x3 + 1) = x1x2x3 + x1x2 + x1x3 + x2x3 + x1 + x2 + x3 + 1

Since P is proper we know that P must have some distinguished facet (this is

evident in the figure). Let f(x) ≥ 0 define this distinguished facet and note that

f must depend on at least two variables. Hence, f 2(x) must have at least one

bilinear term, xixj. However, there is clearly no way to reduce such a term to

the linear level using the generators of I as the only distinguished generator has

leading term x1x2x3 and is therefore too large to be of use in the reduction of a

40


quadratic polynomial. As a result, we know that I cannot be (1, 1)-SOS.

Notice that in the example above it was not necessary to explicitly compute the

facet-defining inequalities for the polytope. The lack of the (1, 1)-SOS property is

evident simply by inspection of the generators of the ideal. Secondly, notice that if

`(x) is any linear polynomial (facet-defining or otherwise) that depends on at least

two variables then `(x) cannot be pseudo-idempotent mod I due to the absence of

a generator with a bilinear leading term. We will highlight this necessary condition

in Proposition 5.2.3. First we explore some additional basic properties of polytopes

that will be of use to us.

5.2 Full-dimensional Polytopes

As we mentioned in Definition 3.2.1, a polytope is the convex hull of a finite set of

points in Rn. These are sometimes called V -polytopes as they are given by a set of

vertices. It is also possible to describe a polytope as the solution set to a system of

linear inequalities, Ax ≤ b. Such polytopes are occasionally called H-polytopes as

they are given by half-spaces. One of the fundamental concepts of polyhedral theory

states that V -polytopes and H-polytopes are equivalent. The interested reader is

referred to [23] for additional information. When a polytope P is given in half-space

form, Ax ≥ b, it is frequently useful to recognize the inequalities in the system that

hold at equality for all points in P . The following notation is often used.

Notation: Let a polytope P be given by the system of inequalities Ax ≥ b. By

(A=, b=) we denote the subsystem of inequalities that hold at equality for all p ∈ P .

Similarly, by (A>, b>) we denote the subsystem of inequalities aix ≥ bi such that

aix0 > bi for some x0 in P .

This gives us a natural way to discuss the dimension of a given polytope. The

following proposition formalizes this notion.

Proposition 5.2.1. Let P ⊆ Rn be a polytope with half-space representation Ax ≤b. Then dim(P ) + rank(A=, b=) = n.

41


See [23] for proof. Much of our work will focus on full-dimensional polytopes. In

this case, we know that rank(A=, b=) = 0. This gives us the following theorem that

we will use frequently.

Theorem 5.2.2. Let P ⊆ Rn be a non-empty binary polytope. Let the correspond-

ing ideal I ⊆ R[x] be given in Grobner basis form for a graded monomial order.

Then P is full-dimensional if and only if I has no linear generators.

Proof. This theorem is easier to approach via the contrapositive. Suppose that

P is not a full dimensional polytope. Then rank(A=, b=) 6= 0. As such, there

must exist a linear polynomial `(x) such that `(p) = 0 for all p ∈ vert(P ). Since

vert(P ) = V(I) and I is radical, we know that ` ∈ I. Thus, some generator

of I has a leading term that divides the linear leading term of `(x). Since P is

non-empty we know I 6= 〈1〉. So any Grobner basis for I must contain a linear

polynomial.

Conversely, suppose that I contains a linear generator `(x). As before, we know

that `(p) = 0 for all p ∈ V(I), and therefore for all p ∈ vert(P ). So ` is a face-

defining inequality for P , (in this case the face is P itself). It follows that some

scalar multiple of `(x) must be present in (A=, b=). By the preceding proposition,

we know that P cannot be full-dimensional. �

The theorem above allows us to make a number of useful claims about the gener-

ating set of (1, 1)-SOS ideals corresponding to full-dimensional polytopes. In partic-

ular, the argument outlined in Example 5.1.8 can now be generalized in the following

proposition.

Proposition 5.2.3. Let I ⊆ R[x] be a binary ideal corresponding to a full-dimensional

proper polytope P ⊆ Rn. If I is (1, 1)-SOS then the reduced Grobner basis for Ihas a distinguished generator with a bilinear leading term.

Proof. Since P is proper we know that P has a distinguished facet given by f(x) ≥0. Furthermore, we know that f(x) cannot be univariate. Consequently, f 2(x)

must contain a bilinear term. Since I is (1, 1)-SOS we know that there exists a

42


linear polynomial `(x) such that f 2 ≡ ` mod I. Since for all i we know that the

square monomials αix2i can be reduced to the constant level by virtue of the x2i −1

generators, we know that f 2(x) must be congruent to some quadratic polynomial

f ′(x) that has leading monomial xixj for i 6= j. Because we also have f ′(x) ≡ `(x)

it follows that I must have some generator g(x) with leading term that divides xixj.

Since P is full-dimensional we know by Theorem 5.2.2 that the leading monomial

of g must be xixj. �

The proposition above allows us to make a claim about the maximum number of

vertices in a proper polytope corresponding to a (1, 1)-SOS ideal. We will make use

of the following relationship between the algebraic notion of standard monomials and

the geometric notion of varieties. We remind the reader of the following definitions.

Definition 5.2.4. Let I ⊆ R[x] be an ideal and suppose a monomial order has

been established. By LT (I) we denote the leading term ideal associated to I. This

is the ideal generated by the monomials in the set

G = {xα | axα is the leading term of some f(x) ∈ I for some a ∈ R}

We will call a monic monomial xα a standard monomial for I if xα /∈ LT (I).

One may regard the standard monomials of a given ideal I as a basis for the

quotient ring R[x]/I as a vector space over R. Notice that if I is zero-dimensional

then I has finitely many standard monomials. The two notions are intimately related

as the following proposition illustrates.

Proposition 5.2.5. Let I ⊆ R[x] be a zero-dimensional ideal. Suppose the I has

k distinct standard monomials for some k ∈ N. Then |V(I)| ≤ k with equality if

and only if I is radical.

The reader is referred to [8] for proof. This relationship between standard mono-

mials and points in the variety allows us to count the vertices of a binary polytope by

inspecting the generators of the corresponding ideal. We obtain the following upper

43


bound for the cardinality of vert(P ) for (1, 1)-SOS ideals.

Theorem 5.2.6. Let P ⊆ Rn be a full-dimensional, proper binary polytope and let

I ⊆ R[x] be the corresponding ideal. If I is (1, 1)-SOS then |vert(P )| ≤ 2n− 2n−2.

Proof. First note that there are 2n binary points in Rn. We remark that there

are also 2n square-free monic monomials in n variables (consider xα where α is a

traditional 0−1 binary n-tuple). Notice that since x2i −1 is present in every binary

ideal for all i, any standard monomial of a binary ideal must be square-free.

Suppose that I is (1, 1)-SOS and is in Grobner basis form for a graded monomial

order. By Proposition 5.2.3 we know that I must have a distinguished generator

of degree 2. Let xixj for i 6= j be the leading monomial of this distinguished

generator. Then any monomial that is divisible by xixj cannot be a standard

monomial of I. Notice that there are 2n−2 square-free monic monomials that are

divisible by xixj in R[x]. Therefore, the maximum number of standard monomials

for I is 2n − 2n−2. Since I is real radical, and thus radical, the result then follows

from Proposition 5.2.5. �

The relationship between standard monomials of binary ideals and the vertices of

the corresponding polytope will also allow us to formalize an intuitive claim about the

minimum number of vertices in a polytope that is not (1, 1)-SOS. The counterpart to

Theorem 5.2.6 follows.

Theorem 5.2.7. In Rn, every full-dimensional simplicial polytope is (1, 1)-SOS.

Proof. Let P ⊆ Rn be a full-dimensional binary polytope with corresponding ideal

I ⊆ R[x]. Suppose that P has n + 1 vertices. Since P is full-dimensional we

know that I cannot contain any linear polynomials. Hence, for i = 1, . . . , n the

monomial xi is a standard monomial for I. Since the constant 1 is a standard

monomial for all non-trivial ideals we see by Proposition 5.2.5 that {1, x1, . . . , xn}are the only standard monomials for I. Since these form a basis for R[x]/I as

an R vector space, we see that all polynomials (facet-defining and otherwise) are

congruent to some linear polynomial in the quotient ring for I. Hence, the facet-

44


defining polynomials for P are pseudo-idempotent in R[x]/I. By Corollary 4.2.2,

we conclude that I is (1, 1)-SOS. �

5.3 The Facet-Generator Correspondence

We have begun exploring the notion that the (1, 1)-SOS property of an ideal hinges

upon the ability to reduce certain quadratic polynomials to the linear level. We have

alluded to the fact that this requires a coordinated arrangement of the bilinear terms

in the generators of the ideal. At the most basic level, Proposition 5.2.3 tells us that,

for proper full-dimensional polytopes, we must have at least one generator with a

bilinear leading term if we wish our ideal to be (1, 1)-SOS. The key concept in the

discussion of this proposition was the fact that a distinguished facet of a polytope

depends on at least two variables. Certainly, if the variables xi and xj appear in a

linear polynomial f(x), then the bilinear monomial xixj appears in f 2(x). If we wish

for f to be idempotent with respect to an ideal I, then we must have some way to

address the xixj monomial using the generators of a Grobner basis of I. Thus, it is

worthwhile for us to develop a theory that illuminates the relationship between the

variables that appear in the facets of a polytope and the variables that appear in the

bilinear terms in the generators of an ideal. We take our first step in this direction

with the following theorem.

Theorem 5.3.1. Let P ⊆ Rn be a full-dimensional, proper binary polytope and let

I ⊆ R[x] be the corresponding ideal. Let xi and xj be distinct variables in R[x]. If

P has a facet of the form f(xi, xj) = aixi + ajxj + a0 ≥ 0 where ai 6= 0 and aj 6= 0

then I contains a quadratic polynomial that properly depends only on xi and xj.

Moreover, if I is in reduced Grobner basis form for a graded monomial order then

this polynomial is a generator.

Proof. Suppose that f(xi, xj) ≥ 0 defines a facet F of P . Consider the four possible

values of the pair (xi, xj) in {−1, 1}2. Since F is distinguished it cannot be a facet

of the hypercube. Thus, there must be some value (pi, pj) ∈ {−1, 1}2 for which

45


f(pi, pj) < 0. Consider the polynomial

ψ(xi, xj) = (xi + pi)(xj + pj)

Let v = (v1, . . . , vn) be a vertex of P . Notice that (vi, vj) 6= (pi, pj). Thus, since v

is binary, we have ψ(v) = 0. Therefore, ψ ∈ I since I is radical.

Suppose that I is in reduced Grobner basis form for a graded monomial order.

Notice that the leading term of ψ is xixj. We know that I has some generator

g(x) such that the leading term of g divides xixj. Also, since P is full dimensional

we know by Theorem 5.2.2 that the leading term of g must be xixj. Consider the

polynomial g−ψ ∈ I. If g−ψ = 0 the proof is complete. Suppose that g−ψ 6= 0.

Again by Theorem 5.2.2 we know that g − ψ cannot be linear. Let xkx` be the

leading monomial of g−ψ. As xixj is the only quadratic monomial in ψ we see that

xkx` must be a term in g. However, since g−ψ lies in I, there must be a generator

of I with leading term that divides xkx`. This contradicts the assumption that Iis in reduced Grobner basis form. Therefore, g = ψ and the proof is complete. �

We exhibit this concept in the following example.

Example 5.3.2. Let P ⊆ R3 be the polytope pictured below.

(1,−1,−1)

(−1,−1, 1)

(1, 1,−1)

(−1, 1,−1)

Figure 5.2: A polytope with bivariate facets

46


We compute the facets of P and obtain

x1 + 1 ≥ 0, x2 + 1 ≥ 0, x3 + 1 ≥ 0

−x2 − x3 ≥ 0, −x1 − x3 ≥ 0

Notice that P contains a facet that depends only on x2 and x3. We also have a

facet that depends only on x1 and x3. We construct the ideal I corresponding to

P using exclusion polynomials.

I = 〈x21 − 1, x22 − 1, x23 − 1, (x1 + 1)(x2 − 1)(x3 + 1),

(x1 + 1)(x2 + 1)(x3 + 1), (x1 − 1)(x2 + 1)(x3 + 1)〉

Computing a Grobner basis for I with respect to graded lexicographic order with

x1 > x2 > x3 we have

I = 〈x21 − 1, x22 − 1, x23 − 1, x1x3 + x1 + x3 + 1, x2x3 + x2 + x3 + 1〉

We emphasize that the Grobner basis for I contains a generator that properly

depends only on x1 and x3 as well as a generator that depends only on x2 and x3.

Theorem 5.3.1 will provide a foundation for much of our later analysis. However,

we warn that the converse of Theorem 5.3.1 does not hold in general. We provide a

brief counterexample.

Example 5.3.3. Let P ⊆ R3 be the polytope pictured below.

The facets of this polytope are

x1 + 1 ≥ 0, x2 + 1 ≥ 0, x3 + 1 ≥ 0, −x1 − x2 − x3 − 1 ≥ 0

However, when we compute the corresponding ideal we obtain

I = 〈x21−1, x22−1, x23−1, x1x2+x1+x2+1, x1x3+x1+x3+1, x2x3+x2+x3+1〉

We see that I contains a generator that properly depends only on x1, x2, for

47


(1,−1,−1)

(−1,−1, 1)

(−1, 1,−1)

Figure 5.3: A polytope without bivariate facets

example, however P does not contain a facet that properly depends on only these

two variables.

While Theorem 5.3.1 will be useful to us, the absence of a converse is unfortunate.

The theorem allows us to inspect the facets of a polytope and make claims about the

variables that appear in the generators of the associated ideal. Overall, the reverse

direction - a theory that provides information about the variables present in facets

based upon those present in the ideal’s generators - will be much more helpful to our

analysis. Luckily, we will be able to construct theorems that operate in this direction

as well. A central notion in this work will involve a form of duality between the facets

of a polytope and the generators of an ideal. In the case of binary polytopes and

ideals, both the facets and generators communicate which binary points are and are

not present in the geometric component of the problem. It is enormously useful to

fluidly translate this information from the form of polynomial equations to the form

of linear polynomial inequalities and vice versa. The first of our steps in this direction

comes with the following lemma.

Lemma 5.3.4. Let P ⊆ Rn be a full-dimensional, proper binary polytope. Let

f(xi, xj) ≥ 0 define a facet of P and suppose that f depends properly on xi and

xj. Given a binary pair p ∈ {−1, 1}2, suppose that f(p) < 0. Then for any

q ∈ {−1, 1}2 with q 6= p we have f(q) ≥ 0.

48


Proof. Suppose by contradiction that f(p) < 0 and f(q) < 0 for distinct points

p, q ∈ {−1, 1}2. Let p = (p1, p2) and q = (q1, q2). Then the polynomials

ψp = (xi + p1)(xj + p2) and ψq = (xi + q1)(xj + q2)

lie in the corresponding ideal I ⊆ R[x]. Hence, the difference ψp − ψq must lie in

I as well. However, if p 6= q then ψp−ψq is a non-zero linear polynomial. Since P

is full-dimensional, we arrive at a contradiction due to Theorem 5.2.2. �

Hence, a facet of a binary polytope that properly depends on two variables is

characterized by the unique binary pair that the facet excludes. We may slightly

generalize this notion using an argument that is nearly identical to the preceding

proof. The result is the following lemma.

Lemma 5.3.5. Let P ⊆ Rn be a full-dimensional, proper binary polytope. Given

a pair of distinct variables xi and xj, P can have at most one facet that properly

depends only on xi and xj.

Proof. Suppose that P has facets F given by f(xi, xj) ≥ 0 and G given by

g(xi, xj) ≥ 0. Suppose that f and g properly depend on xi and xj. We know

that for some binary pairs p, q ∈ {−1, 1}2 we have f(p) < 0 and g(q) < 0. Let

p = (p1, p2) and q = (q1, q2). Then the polynomials

ψp = (xi + p1)(xj + p2) and ψq = (xi + q1)(xj + q2)

must lie in the corresponding ideal I ⊆ R[x]. However, this means that ψp−ψq ∈ I.

Notice that if ψp − ψq 6= 0 then it must be a linear polynomial. Since P is full

dimensional, we know by Theorem 5.2.2 that ψp − ψq = 0. Therefore, p = q and,

by Lemma 5.3.4, we conclude that f and g define the same facet of P . �

These lemmas will enable us to make a statement about which variables appear in

the facets of a polytope based upon the variables in the generators of the associated

ideal. As we previously mentioned, this is the direction that will be most helpful to

us. The theorem follows after a couple of observations regarding Buchberger’s algo-

49


rithm.

We introduce the following notation

Notation: Let f ∈ R[x] be a polynomial. With respect to an established mono-

mial order we will use LT (f) to denote the leading term of f .

Recall the notion of the S-polynomial corresponding to a given pair of polynomials

(sometimes called Syzygy pairs or critical pairs). These are the polynomials defined

as follows.

Definition 5.3.6. Let f, g ∈ R[x] be non-zero polynomials. We denote the least

common multiple of the leading terms of f and g by M = lcm(LT (f), LT (g)).

Then the S-polynomial for f and g is given by

S(f, g) =M

LT (f)f − M

LT (g)g

The cornerstone of the theory of Grobner bases is Buchberger’s theorem. We remind

the reader of the statement of this theorem.

Theorem 5.3.7 (Buchberger’s Theorem). Let G = {g1, . . . , gk} ⊆ R[x] be a set

of polynomials. Let I = 〈g1, . . . , gk〉. Then G is a Grobner basis for I if and

only if for all i 6= j the polynomial S(gi, gj) reduces to zero upon division by the

polynomials in G (in any order).

For the proof, as well as a thorough and accessible tour of the applications of

Grobner bases, we refer to [1]. In our study we will encounter situations in which

we wish to make claims regarding the appearance of the generators of our ideals (in

fact, we already have). However, the generating set of an ideal is mutable unless we

assume we’re operating with a reduced Grobner basis. As such, it will be necessary

to explore some principles that govern the form of Grobner bases. We accomplish

this with a pair of technical lemmas.

50


Lemma 5.3.8. Let G = {g1, . . . , gk} ⊆ R[x] be a collection of polynomials. If

LT (gi) and LT (gj) have no variables in common then S(gi, gj) reduces to zero

upon division by G.

For proof we refer to Buchberger’s paper [5]. We also establish the following claim.

Lemma 5.3.9. For i = 1, . . . , k we consider pairs of variables (xi1, xi2) ∈ {x1, . . . , xn}2

where i1 6= i2. Also for i = 1, . . . , k let pi = (pi1, pi2) ∈ {−1, 1}2 be binary pairs.

We define the polynomials ψi ∈ R[x] as follows

ψi = (xi1 + pi1)(xi2 + pi2)

Then the ideal given by

I = 〈x21 − 1, . . . , x2n − 1, ψ1, . . . , ψk〉

is in Grobner basis form for graded lexicographic order with x1 > · · · > xn.

Proof. We will show that the S-polynomial associated to each pair of generators

of I reduces to zero upon division by the generators of I as listed above. Then,

by Theorem 5.3.7, the proof will be complete. Assume that i 6= j throughout.

First note that we need not consider the polynomials S(x2i − 1, x2j − 1) due to

Lemma 5.3.8. Similarly, if xi1, xi2, xj1 and xj2 are all distinct then we need not

bother with S(ψi, ψj) by Lemma 5.3.8. We suppose that (xi2 + pi2) = (xj2 + pj2)

and we consider

ψi = (xi1 + pi1)(xi2 + pi2) and ψj = (xj1 + pj1)(xi2 + pi2)

Notice

(xj1 + pj1)ψi = (xi1 + pi1)ψj

and so

S(ψi, ψj) = xj1ψi − xi1ψj = pi1ψj − pj1ψi

Clearly, since LT (ψi) 6= LT (ψj), this polynomial reduces to zero upon division by

51


ψi and ψj.

Next we consider the pair x2i − 1 and ψj and we assume that xj1 = xi. We have

S(x2i − 1, (xi + pj1)(xj2 + pj2)) = xj2(x2i − 1)− xi(xi + pj1)(xj2 + pj2)

= −pj1pj2xi − pj1xixj2 − pj2x2i − xj2

Notice that we may replace −pj2x2i with −pj2. Regarding the remaining terms,

suppose that pj1 = 1. Then our S-polynomial is

−pj2xi − xixj2 − pj2 − xj2 = −(xi + 1)(xj2 + pj2) = −ψj

Likewise, if pj1 = −1 we have

pj2xi + xixj2 − pj2 − xj2 = (xi − 1)(xj2 + pj2) = ψj

In either case, the S-polynomial clearly reduces to zero by virtue of the ψj generator

alone. Therefore, by Theorem 5.3.7 we conclude that I is in Grobner basis form.

�

The lemma above can actually be significantly strengthened as the choice of the

monomial order need not be specified. However, this is peripheral to our purposes

and we will not explore it further. We have now established the necessary groundwork

to justify the following claim.

Theorem 5.3.10. Let P ⊆ Rn be a full-dimensional, proper binary polytope. Let

I ⊆ R[x] be the ideal corresponding to P in reduced Grobner basis form for a

graded monomial order. Suppose that all distinguished facets of P properly depend

on exactly two variables each. Then all distinguished generators of I properly

depend on exactly two variables each.

Proof. Let P have distinguished facets F1, . . . , Fk. Let each facet Fi be given by

fi(xi1, xi2) ≥ 0. We assume that each fi properly depends on xi1 and xi2. Moreover,

we know by Lemma 5.3.4 that each Fi corresponds to a unique binary pair pi =

(pi1, pi2) ∈ {−1, 1}2 with fi(pi) < 0. For i = 1, . . . , k we define ψi(x) ∈ R[x] as

52


follows

ψi(x) = (xi1 + pi1)(xi2 + pi2)

We then construct the following ideal.

I ′ = 〈x21 − 1, . . . , x2n − 1, ψ1, . . . , ψk〉

Notice that by Lemma 5.3.9 we know that I ′ is in Grobner basis form for a graded

monomial order. We claim that I ′ = I. Let q = (q1, . . . , qn) ∈ {−1, 1}n. Suppose

that q ∈ vert(P ). Then fi(q) ≥ 0. This means that ψi(q) = 0. Therefore, q ∈ V(I).

Similarly, if q ∈ V(I) we know that ψ(q) = 0 for all i. Thus, fi(q) ≥ 0 for all i.

Consequently, we know that q ∈ P . By Lemma 4.1.2 we know that q ∈ vert(P ).

Therefore, R√I ′ = I. However, for arguments identical to those presented in the

proof of Lemma 4.3.4, we know that I ′ is real radical. Therefore, I ′ = I and the

proof is complete. �

This yields the following claim.

Corollary 5.3.11. Suppose that P ⊆ Rn is a full-dimensional, proper binary

polytope. Let I ⊆ R[x] be the ideal corresponding to P in reduced Grobner basis

form for a graded monomial order. If I has a generator that properly depends on

three or more variables then P has a facet that properly depends on three or more

variables.

Proof. This is the contrapositive of Theorem 5.3.10. �

We emphasize that the above corollary operates in the generators → facets direc-

tion. This is immensely useful as the following example illustrates.

Example 5.3.12. Consider the following binary ideal I in R[x1, x2, x3] and note

that I is in Grobner basis form.

I = 〈x21 − 1, x22 − 1, x23 − 1, x1x3 + x2x3 + x1 + x2〉

53


Without the need for any computation at all we can claim that I is not (1, 1)-SOS.

Notice the distinguished generator x1x3 + x2x3 + x1 + x2. By Corollary 5.3.11 we

know that the corresponding polytope P ⊆ R3 must have a facet that depends on

all three of our variables. Let f(x1, x2, x3) ≥ 0 be this facet. Then each of the three

bilinear terms xixj must appear in f 2. However, notice that I does not contain

a generator that includes the monomial x1x2, nor are there any linear generators.

Consequently, there is no way to reduce f 2 to the linear level via division by the

generators of I. As a result, I cannot be (1, 1)-SOS.

We compare this to the existing method for verification of the (1, 1)-SOS property.

One can compute the points in VR(I) and find the convex hull. The result is the

polytope P ⊆ R3 pictured below.

(1,−1,−1)

(1, 1,−1)

(−1, 1,−1)

(1,−1, 1)

(−1, 1, 1)

Figure 5.4: The polytope corresponding to I

The facets of this polytope are given by

x1 + 1 ≥ 0, x2 + 1 ≥ 0, x3 + 1 ≥ 0, −x1 + 1 ≥ 0, −x2 + 1 ≥ 0

x1 + x2 − x3 + 1 ≥ 0, −x1 − x2 − x3 + 1 ≥ 0

Notice that, as expected, P contains facets that properly depend on all three vari-

ables. One may proceed to show that these distinguished facets are not idempotent

in the quotient ring R[x]/I.

54


Bivariate facets and generators are going to be exceptionally helpful to our analysis

of the (1, 1)-SOS property. The following proposition demonstrates that the bivariate

facets will always be cooperative with sum of squares techniques.

Proposition 5.3.13. All facets of binary polytopes that are given by polynomials

`(xi, xj) that properly depend on exactly two distinct variables are compliant.

Proof. Let P ⊆ Rn be a binary polytope and let I ⊆ R[x] be the corresponding

ideal. Suppose there exists a facet F of P that is given by the linear inequality

`(xi, xj) ≥ 0 where ` properly depends on xi and xj. By Lemma 5.3.4 we know

that there exists a unique point (pi, pj) ∈ {±1}2 such that `(pi, pj) < 0. So the

polynomial ψ(xi, xj) = (xi+pi)(xj+pj) vanishes on all vertices of P . Hence, ψ ∈ I.

Since xixj is the only bilinear monomial in both ψ and `2, we conclude that ` must

be pseudo-idempotent in R[x]/I. Therefore, F is a compliant facet. �

The proposition above shows us that a bivariate facet will never violate the (1, 1)-

SOS property. We emphasize that the proposition holds even when the polytopes

in question are not full-dimensional. We remark that in R3 any polytope that has a

facet that depends on exactly two variables will be (1, 1)-SOS. This, of course, will

not hold in general, but it can be a helpful shortcut for small problems.

The correspondence between the facets of a polytope and the generators of the

corresponding ideal is of critical importance to our study. As discussed above, the

claims that allow us to anticipate the structure of the facets of a polytope given the

generators of the corresponding ideal are of particular value to us. However, there

are a number of counterexamples that frustrate many of the tempting claims that

operate in this direction (see the following chapter on counter-examples for details).

As such, many of our instrumental claims will arise as contrapositives of theorems

that operate in the more cooperative facets → generators direction. The preceding

Corollary 5.3.11 is one such claim. We offer another below.

Theorem 5.3.14. Let P ⊂ Rn be a full-dimensional proper binary polytope. Let

I ⊂ R[x] be the corresponding ideal. Given a variable xj ∈ {x1, . . . , xn}, suppose

55


that no distinguished facet of P is given by a polynomial that properly depends on

xj. Then no distinguished generator of I properly depends on xj.

Proof. Suppose that the variable xj does not appear in the polynomial defining

any distinguished facet of P . Let (p1, . . . , pj, . . . , pn) ∈ {−1, 1}n be a binary point

that is not a vertex of P . Notice that (p1, . . . ,−pj, . . . , pn) cannot be a vertex of

P either. Consequently, the polynomial

ψ = (x1 + p1) · · · (xj−1 + pj−1)(xj+1 + pj+1) · · · (xn + pn)

must vanish on all vertices of P . Therefore, ψ ∈ I. For each binary point p that

is absent from vert(P ) we may construct such a polynomial ψp. Let ψ1, . . . , ψk be

the (possibly redundant) result. Consider the ideal

I ′ = 〈x21 − 1, . . . , x2n − 1, ψ1, . . . , ψk〉

Notice that by Lemma 4.3.4 we know that I ′ is real radical and that convV(I ′) = P .

Since the real radical ideal corresponding to a given binary polytope is unique, we

see that I ′ = I. Now consider the application of Buchberger’s algorithm to the

polynomials generating I ′. Given ψi1 and ψi2 we know that the variable xj does

not appear in either polynomial. In particular, xj does not appear in the leading

monomial for either ψi1 or ψi2. Consequently, xj cannot be introduced in the

S-polynomial for ψi1 and ψi2. As a result, the variable xj cannot appear in the

reduced Grobner basis of the set

S = {x21 − 1, . . . , x2j−1 − 1, x2j+1 − 1, . . . , x2n − 1, ψ1, . . . , ψk}

Let G denote the reduced Grobner basis of S. Then, by Lemma 5.3.8, we know

that the Grobner basis of the set S ∪ {x2j − 1} must be equal to G ∪ {x2j − 1}.Therefore, xj cannot appear in any of the distinguished generators of the reduced

Grobner basis for I. �

The contrapositive of the preceding theorem will allow us to achieve a useful claim

for the algebraic characterization of the (1, 1)-SOS property. We state this below.

56


Corollary 5.3.15. Let P ⊆ Rn be a full-dimensional proper binary polytope and

let I ⊆ R[x] be the corresponding ideal. If a variable xj appears in a distinguished

generator of I then xj appears in some distinguished facet of P .

Proof. This is the contrapositive of Theorem 5.3.14. �

5.4 Sums of Ideals

We close this chapter with an exploration of the behavior of (1, 1)-SOS ideals under

sums. Certainly, it is possible to add ideals that are not (1, 1)-SOS to achieve a

(1, 1)-SOS ideal. However, the question of whether or not the sum of (1, 1)-SOS

ideals remains (1, 1)-SOS is considerably more interesting. For example, consider the

polytope in R4 with corresponding ideal

I = 〈x21−1, x22−1, x23−1, x24−1, x1x2+x1+x2+1, x2x4−x3x4+x2−x3, x2x3+x2−x3−1〉

We may regard I as the sum of ideals I1 and I2 given by

I1 = 〈x21 − 1, x22 − 1, x1x2 + x1 + x2 + 1〉

and

I2 = 〈x22 − 1, x23 − 1, x24 − 1, x2x4 − x3x4 + x2 − x3, x2x3 + x2 − x3 − 1〉

We may be sure that I1 is (1, 1)-SOS due to the compliance of facets on two variables.

Additionally, notice that I2 is isometric to the polytope discussed in Example 3.3.2

and thus must be (1, 1)-SOS. Are these observations sufficient to claim that I =

I1 + I2 is (1, 1)-SOS? The answer is more nuanced than one may initially expect.

Our exploration will be assisted by the following lemma.

Lemma 5.4.1. Let P1 and P2 be full-dimensional binary facets in Rn. Suppose

that dim(P1 ∩ P2) = n. Then each facet-defining inequality of the polytope P1 ∩ P2

is a facet-defining inequality for either P1 or P2 (or both).

57


Proof. Suppose that P1 and P2 are given in half-space representation by the sys-

tems A1x ≥ b1 and A2x ≥ b2. Recall that in the discussion of Proposition 5.2.1

we introduced the notation (A>,b>) to denote the subsystem of inequalities in

Ax ≥ b that hold at strict equality for some point of the corresponding polytope.

Let P = P1 ∩ P2 and suppose that dim(P ) = n. Consider the system Ax ≥ b

where

A =

(A1

A2

)and b =

(b1

b2

)This system of inequalities yields the polytope P . By Proposition 5.2.1 we know

that (A=1 ,b1

=) and (A=2 ,b2

=) must be empty. Likewise, we know that since P

is full-dimensional we must have (A=,b=) empty as well. Hence, every inequal-

ity in Ax ≥ b must be present in (A>,b>). One of the fundamental theorems

of optimization over polyhedra states that the facet-defining inequalities of P are

necessary and sufficient in (A>,b>). See [23,29]. Consequently, every facet of P is

given by some inequality in the system Ax ≥ b. As this system was constructed

by the concatenation of the half-space representations of P1 and P2 we know that

each facet-defining inequality for P is present in either A1x ≥ b1 or A2x ≥ b2.

Since, for i = 1, 2, the facet-defining inequalities for Pi are necessary and sufficient

in (A>i ,bi>) = (Ai,bi), we may assume without loss of generality that every in-

equality in the half-space representation of each Pi is facet-defining. Consequently,

every facet of P is given by a facet-defining inequality that defines a facet of either

P1 or P2. �

This lemma will enable us to claim that the (1, 1)-SOS property is preserved under

sums for ideals that are “cooperative” in a sense that we presently illuminate. To be

thorough, we bolster our claim with an additional observation.

Lemma 5.4.2. Let P1 and P2 be full-dimensional binary polytopes in Rn with

corresponding ideals I1 and I2 respectively. Let P = P1 ∩ P2 be a full-dimensional

binary polytope in Rn. Then I1 + I2 is the ideal corresponding to P .

Proof. Recall that the sum of ideals is the algebraic counterpart to the geometric

notion of the intersection of varieties. That is, I1+I2 has the variety V(I1)∩V(I2).

58


By Lemma 4.1.2 we know that since P = P1 ∩ P2 is binary we have vert(P ) =

vert(P1)∩vert(P2). Consequently, vert(P ) = V(I1+I2) = VR(I1+I2). Since I1+I2is radical we conclude that I1 + I2 is the unique real radical ideal corresponding

to P1 ∩ P2. �

It is worthwhile to note that the proof presented above relies upon the fact that

the sum of real radical binary ideals remains real radical. We warn that, in general,

the sum of radical ideals need not be radical. For example, the ideals I1 = 〈x〉 and

I2 = 〈x+ y2〉 are radical ideals, however

I1 + I2 = 〈x, y2〉 6= 〈x, y〉 =√〈x, y2〉

In the proof of Lemma 5.4.2 we are safe, however, since we restrict our attention to

binary ideals. In this case, the radical property of our ideals is maintained across sums

since the square-free univariate generators are not lost. This is the same argument

employed in the proof of Lemma 4.3.4. These lemmas justify the following claim.

Theorem 5.4.3. Let P1 and P2 be full-dimensional binary polytopes in Rn with cor-

responding ideals I1 and I2 respectively. Suppose that P1∩P2 is a full-dimensional

binary polytope. If I1 and I2 are (1, 1)-SOS then I1 + I2 is (1, 1)-SOS.

Proof. Since P = P1 ∩ P2 is full-dimensional we know by Lemma 5.4.1 that every

facet-defining inequality of P must also define a facet of either P1 or P2. Addi-

tionally, since P is binary we know by Lemma 5.4.2 that I1 + I2 must be the real

radical ideal corresponding to P . Let `(x) ≥ 0 define a facet of P . Since both

I1 and I2 are (1, 1)-SOS we know that `(x) must be pseudo-idempotent in either

R[x]/I1 or R[x]/I2. In either case, we may be sure that `(x) is pseudo-idempotent

in R[x]/(I1 + I2). Therefore, I1 + I2 is (1, 1)-SOS. �

In much of our study thusfar we have treated our ideals and polytopes as somewhat

interchangeable algebraic and geometric objects. This attitude is reflected through-

out much of the literature on polyhedral optimization. Certainly any polytope is

uniquely determined by its set of vertices and in many discussions it is not necessary

59


to emphatically distinguish between polytopes and associated vertex sets. However,

the study of sums of (1, 1)-SOS ideals is significantly influenced by this distinction.

Our ideals define the vertex sets of their corresponding polytopes. However, given

polytopes P1 and P2 the polytope P1 ∩ P2 is not necessarily equal to the polytope

conv(vert(P1) ∩ vert(P2)). As such, the polytope corresponding to I1 + I2 need not

be equal to P1 ∩ P2. The trouble lies in the fact that P1 ∩ P2 may have vertices that

are not binary points. This important notion is illustrated in the following example.

Example 5.4.4. Consider the polytopes P1 and P2 in R3 pictured below.

(1,−1,−1)

(−1,−1, 1)

(−1, 1,−1)

(1, 1,−1) (1,−1,−1)(−1, 1,−1)

(1,−1, 1)

(−1,−1, 1)

(−1, 1, 1)

Figure 5.5: Two binary polytopes in R3

The left polytope corresponds to the ideal

I1 = 〈x21 − 1, x22 − 1, x23 − 1, x1x3 + x1 + x3 + 1, x2x3 + x2 + x3 + 1〉

and the right polytope gives us

I2 = 〈x21 − 1, x22 − 1, x23 − 1, x1x2 + x1 + x2 + 1〉

The following figure illustrates an important distinction.

60


(1,−1,−1)

(−1,−1, 1)

(−1, 1,−1)(1,−1,−1)

(−1,−1, 1)

(−1, 1,−1)

(0, 0, 0)

Figure 5.6: Sums of Ideals vs Intersections of Polytopes

The polytope on the left is the polytope corresponding to the sum I1 + I2.The right polytope is the result of the intersection P1∩P2. We emphasize that the

non-binary point (0, 0, 0) is a vertex of the polytope obtained from the intersection.

Hence, I1 + I2 is not the ideal corresponding to the polytope P1 ∩ P2.

This phenomenon can greatly complicate the study of the sums of (1, 1)-SOS

ideals. In the example above the polytope corresponding to I1 + I2 is indeed (1, 1)-

SOS (see Example 5.3.3). However, the facet defined by −x1− x2− x3− 1 ≥ 0 is not

a facet of either of the polytopes P1 and P2. In general, it is not so obvious that the

linear polynomials that define these “new” facets will always be pseudo-idempotent

with respect to I1 + I2. We are inspired to explore the nature of the new facets that

we may encounter when studying the sum of ideals. We will use the following result

from the geometric analysis in [10].

Proposition 5.4.5. Let P be a full-dimensional binary polytope in Rn and let

I ⊆ R[x] be the corresponding ideal. Then I is (1, 1)-SOS if and only if every facet

of P is given by a facet-defining inequality `(x) ≥ 0 that satisfies `(p) ∈ {0, 1} for

all p ∈ V(I) = vert(P ).

Proof. We know that I is (1, 1)-SOS if and only if every facet can be given by

a linear polynomial that is idempotent in R[x]/I. However, a polynomial `(x)

satisfies `2(x) ≡ `(x) mod I if and only if `(x)(`(x)− 1) ≡ 0 mod I. Hence, over

the variety V(I) we know that `(x) may only take on the values 0 and 1. �

61


We can slightly relax the stipulation in this proposition. If a facet is given by an

inequality `(x) ≥ 0 that satisfies `(p) ∈ {0, α} for all p ∈ vert(P ) where α ∈ R, then

we can always scale ` to achieve an equivalent representation `′(x) ≥ 0 that yields

`′(p) ∈ {0, 1} for all p ∈ vert(P ). Overall, for a facet F given by `(x) ≥ 0 it is sufficient

that ` map all vertices of P that do not lie on F to the same value. Geometrically,

this states that all vertices of P either lie on F or on a unique hyperplane that is

parallel to F .

We will offer a slight extension of Theorem 5.4.3 based on the observations illus-

trated in Example 5.4.4. This will focus on the scenario in which we have a facet of

the polytope corresponding to I1 + I2 that is not a facet of P1 ∩ P2. As in the ex-

ample, we will consider the case in which this facet lies opposite a non-binary vertex

produced by the intersection. We formalize this with a definition.

Definition 5.4.6. Let P1 and P2 be full-dimensional binary polytopes in Rn.

Suppose that P = P1∩P2 is a full-dimensional polytope with a vertex v /∈ {−1, 1}n.

Suppose that the polytope Q = conv(vert(P )\{v}) has exactly one facet F given

by `(x) ≥ 0 where `(x) does not define any facet of P . If every facet of P that

passes through v shares n− 1 vertices with F then we say that F lies opposite v.

So in Example 5.4.4 the facet given by −x1 − x2 − x3 − 1 ≥ 0 lies opposite the

origin. We now offer the following theorem.

Theorem 5.4.7. For n ≥ 3 let P1 and P2 be full-dimensional binary polytopes in

Rn with corresponding (1, 1)-SOS ideals I1 and I2 respectively. Let P = P1 ∩ P2

and let Q be the polytope corresponding to I1 + I1. Let dim(P ) = dim(Q) = n and

suppose that Q has exactly one facet F such that F is given by an inequality that

does not define any facet of P . Let F lie opposite a non-binary point v in P . For

i = 1, . . . , k let ì(x) ≥ 0 define the distinct facets in P that satisfy ì(v) = 0. If

for every pair of vertices q1, q2 of Q with q1, q2 /∈ F we have∑

i ì(q1) =∑

i ì(q2)

then I1 + I2 is (1, 1)-SOS.

Proof. By Proposition 5.4.5 we may assume without loss of generality that the

inequalities ì(x) ≥ 0 define distinct facets of P and satisfy ì(p) ∈ {0, 1} for

62


all vertices p ∈ P . Let f(x) =∑

i ì(x) and notice that there must exist some

constant α ∈ Z such that f(v) = α for all vertices v ∈ F . Hence, f(x) − α must

define the facet F . By Lemma 4.1.2 we know that every vertex of Q must be a

vertex of P . Hence, for all i we know ì(q) ∈ {0, 1} for all vertices of Q. Since

by hypothesis we have∑

i ì(q1) =∑

i ì(q2) for all vertices q1 and q2 of Q that

do not lie on F , we know that there must exist some constant β ∈ Z such that

f(q) = β for all vertices q of Q with q /∈ F . Consequently, the linear polynomial

f ′(x) = f(x)− α satisfies f ′(q) ∈ {0, β − α}. Moreover, since Q is assumed to be

full-dimensional we may assume that β 6= α. Hence, scaling by (β − α)−1 we may

construct a linear polynomial f ′′(x) that satisfies f ′′(q) ∈ {0, 1} for all vertices q of

Q and f ′′(x) ≥ 0 is a defining inequality for F . As all other facets of Q are facets

of P and P is full-dimensional, we know by Lemma 5.4.1 and Theorem 5.4.3 that

there exist facet-defining inequalities gi(x) ≥ 0 for each of the remaining facets of

Q that satisfy gi(p) ∈ {0, 1} for all vertices p ∈ P . Since vert(Q) ⊂ vert(P ), the

proof is complete. �

63


Chapter 6

Necessary Conditions via IP Feasibility

The observations in the previous chapter are motivated by the fact that every zero-

dimensional (1, 1)-SOS ideal corresponds to a polytope that can be constructed with

facets given by pseudo-idempotent linear polynomials. It turns out that the quotient

rings for many binary ideals do not admit any pseudo-idempotent distinguished facet-

defining linear polynomials whatsoever. The ability to quickly recognize such ideals

will prove valuable to our study. We motivate the development of this necessary

condition with an illustration.

Example 6.0.8. Consider the binary polytope in R3 pictured below.

(1, 1,−1)

(−1, 1,−1)

(1,−1, 1)

(−1, 1, 1)

(−1,−1,−1)

Figure 6.1: An illustration of necessary conditions

Using the elimination polynomial approach, we can construct the corresponding

binary ideal. Computing a Grobner basis we obtain

I = 〈x21 − 1, x22 − 1, x23 − 1, x1x3 + x2x3 + x1 + x2, x1x2 − x2x3 − x1 + x3〉

We now study the notion of pseudo-idempotence in the quotient ring R[x]/I.

64


Consider the general affine polynomial

`(x) = ax1 + bx2 + cx3 + d

where a, b, c, d ∈ R. Using Maple we symbolically compute the normal form of `2

with respect to I (we use graded lex order with x1 > x2 > x3). We obtain

`2 ≡ (2ab− 2ac+ 2bc)x2x3 + lower degree terms

Hence, if we wish for `2 to be congruent to any linear polynomial in R[x]/I we

must have

2ab− 2ac+ 2bc = 0

As the example above illustrates, we can formulate a necessary condition for the

(1, 1)-SOS property in terms of the interplay among the coefficients of a general facet-

defining polynomial. Let us examine the condition

2ab− 2ac+ 2bc = 0

further. Of course, a point of the form (a, b, c) = (a, 0, 0) yields a valid solution.

However, any proper binary polytope must have at least one distinguished generator.

As such, we may freely assume that at least two of the coefficients in `(x) are non-

zero. We still have plenty of solutions. For example, (a, b, c) = (6, 2, 3) will suffice.

We are inspired to ask, “Can `(x) = 6x1 + 2x2 + 3x3 + d define a facet of a binary

polytope P ⊆ R3?” As one might have guessed, the answer is no. Though this may be

intuitive in R3, the possible values for the coefficients of the facet-defining hyperplanes

of binary polytopes in higher dimensions is not so obvious. We will explore this notion

further with the help of Hadamard matrices.

6.1 Hadamard Bounds

A facet-defining hyperplane of any binary polytope P in Rn must pass through at

least n binary points in {±1}n. Consider the case of n = 3 and let p = (p1, p2, p3),

q = (q1, q2, q3), and r = (r1, r2, r3) be binary points. Let `(x) = ax1 + bx2 + cx3 +d be

65


the linear polynomial passing through p, q, and r. Recall from basic vector calculus

that we may compute a normal vector for ` (and hence the values of a, b, and c) using

the determinant of the matrix e1 e2 e3

p1 − r1 p2 − r2 p3 − r3q1 − r1 q2 − r2 q3 − r3

where the ei represent the standard basis vectors in R3 (treated symbolically for the

purposes of computing the determinant).

We make some simple yet important observations. Notice that, given a facet F of

a binary polytope P , we may apply an isometry to obtain a polytope P ′ with facet

F ′ corresponding to F such that F ′ passes through the point (−1,−1,−1, . . . ,−1).

In the discussion hereafter we will assume without loss of generality that our facet-

defining hyperplane passes through the point (−1,−1,−1, . . . ,−1). Then, given any

binary point p ∈ {±1}n, the components of the vector 〈p1−(−1), . . . , pn−(−1)〉 must

lie in {0, 2}. After appropriate scaling by 12, we may assume the vector formed by the

points p and (−1,−1, . . . ,−1) has traditional 0−1 binary components. Consequently,

the coefficient of each standard basis vector ei, and therefore the coefficient of the xi

term in the corresponding linear polynomial, may be regarded as the determinant of

a 0 − 1 binary matrix. We are motivated to ask, “What are the possible values for

the determinant of an (n − 1) × (n − 1) matrix with 0 − 1 binary entries?” This is

one of the central questions in the theory of Hadamard matrices.

In the case of R3 it is not difficult to see that the facet-defining hyperplanes of bi-

nary polytopes must have normal vectors that, scaling as necessary, have components

that lie in {−1, 0, 1}. Indeed, consider the determinant of the matrix e1 e2 e3

α1 α2 α3

β1 β2 β3

Applying an isometry as needed, we may assume that our hyperplane passes through

(−1,−1,−1). Hence, without loss of generality we may assume that each αi and βi

66


lies in {0, 1}. Without much trouble one may exhaustively show that the determinant

of a 0 − 1 binary 2 × 2 matrix must lie in {−1, 0, 1}. The coefficients a1, a2, and a3

of a facet-defining linear polynomial `(x) = a1x1 + a2x2 + a3x3 + a0 passing through

(−1,−1,−1) must therefore lie in {−1, 0, 1}. Certainly, the application of a signed

permutation of variables cannot change the ratio of these values. Hence, we see that

any facet-defining linear polynomial for a binary polytope in R3 must have coefficients

that (perhaps after scaling) lie in {−1, 0, 1}.

Can we expect this behavior to hold in higher dimensions? It is not difficult to see

that the answer is no. We can construct counter-examples as early as R4. Suppose

we have a polytope in R4 with a facet F that passes through the points

(1,−1, 1, 1), (1, 1,−1, 1), (1, 1, 1,−1), and (−1,−1,−1,−1)

Let `(x) = a1x1 + a2x2 + a3x3 + a4x4 + a0 be a linear polynomial that defines this

facet. We may obtain the coefficients for ` by computing a normal vector of F . We

construct the matrix e1 e2 e3 e4

1 0 1 1

1 1 0 1

1 1 1 0

The determinant of this matrix yields the normal vector 〈2,−1,−1,−1〉. Hence, F

is given by a polynomial of the form `(x) = 2x1 − x2 − x3 − x4 + a0. The important

observation is that the coefficients of ` do not lie in {−1, 0, 1}. However, research

in Hadamard matrix theory will allow us to safely claim that the coefficients of a

facet-defining polynomial in R4 must lie in {0,±1,±2} after appropriate scaling. We

ultimately seek to develop a necessary condition for the (1, 1)-SOS property of ideals

in terms of feasibility of integer programs. As such, a notion of exactly which integer

values are permissible for the coefficients of facet-defining linear polynomials will be

of immense value to us.

The study of the maximum value of the determinant of matrices of a given type

dates back as early as the 1890’s with the work of J. Hadamard [12]. Researchers in

this field often restrict their focus to 0− 1 binary matrices, ±1 binary matrices, and

67


matrices with entries in {−1, 0, 1}. For our purposes we will utilize results regarding

the value of the determinant of 0− 1 binary matrices. To begin, we note that if A is

a 0− 1 binary n× n matrix then we have

|det(A)| ≤ (n+ 1)(n+1)

2

2n

as an upper bound for the absolute value of our determinant [4, 19]. We will refer

to this value as the Hadamard bound. It is interesting to note that the Hadamard

bound may not hold at equality for certain values of n. Moreover, the existence of a

0− 1 binary n× n matrix with determinant k ∈ N is no guarantee that there exists

such a matrix with determinant k− 1. For example, in [22] it is shown that for 0− 1

binary matrices of size 7× 7 the set of positive determinant values is

{0, 1, 2, . . . , 17, 18, 20, 24, 32}

Notice that, although the Hadamard bound holds at equality, there are gaps in

the sequence! The study of the distribution of positive determinant values of binary

matrices is an active area of research with a number of exciting open problems [24].

The interested reader is encouraged to visit the homepage of Prof. W. Orrick for a

wealth of information about this and other related topics.

We will make use of many of these results in the construction of our integer pro-

grams. The following notation will be useful to us.

Notation: Given a positive integer n we define M(n) as follows:

M(n) = {k ∈ Z | There exists a 0− 1 binary n× n matrix A with det(A) = k}

We emphasize that M(n) will contain both positive and negative integers for n ≥ 1.

The values in M(n) will provide a helpful means of restricting the solutions to the

integer programs that we construct in the following section.

68


6.2 Integer Programs

To begin our formulation of a necessary condition for the (1, 1)-SOS property we will

explore additional notions regarding the permissible values of coefficients for facet-

defining linear polynomials. This is best illustrated with an example.

Example 6.2.1. Suppose that P is a full-dimensional binary polytope in R4. In

the preceding subsection we saw that 〈2,−1,−1,−1〉 is a valid normal vector for

a facet of a binary polytope in R4. We will argue that 〈2, 2, 2, 1〉, or any signed

permutation thereof, cannot be the normal vector for a facet of P .

Suppose by contradiction that v = 〈2, 2, 2, 1〉 is a normal vector for a facet F of P .

Then we may apply an isometry A such that the image of F under A passes through

the point (−1,−1,−1,−1). Let v′ = 〈α1, α2, α3, α4〉 be the image of v under this

isometry. By a simple pigeonhole argument we may claim that there exist two

distinct components αi1 and αi2 of v′ such that αi1 = αi2 and |αi1| = |αi2| = 2.

We also know that there exists a 4× 4 matrix satisfying

det

e1 e2 e3 e4

β21 β22 · · · β24...

...

β41 · · · · · · β44

= 〈α1, α2, α3, α4〉

where βij ∈ {0, 1} for all i, j. Consider replacing the entries of the top row of the

matrix above by 0 − 1 binary values where ei1 = ei2 = 1 and ej = 0 otherwise.

The determinant of the resulting matrix is therefore αi1 + αi2 = ±4. However,

the Hadamard bound for n = 4 is roughly 3.5 (note that, as binary matrices must

have integer determinants, we will often round such figures down). Consequently,

there cannot exist a 0 − 1 binary 4 × 4 matrix with determinant ±4. Hence,

neither 〈2, 2, 2, 1〉 nor any signed permutation thereof can be a normal vector for

any binary polytope in R4.

As the example above illustrates, we can employ the Hadamard bound to achieve

restrictions not only on the value of the components of the normal vectors for our

facets, but on the sums and differences of those values. We will generalize this idea

69


with a theorem momentarily. We first establish a definition that will ensure that the

components of our vectors are not unnecessarily large.

Definition 6.2.2. Let v = 〈α1, . . . , αn〉 be a vector. We will say that v is integrally

scaled if αi ∈ Z for all i = 1, . . . , n and the greatest common factor of {α1, . . . , αn}is 1.

We are now prepared to generalize the notion illustrated in Example 6.2.1

Theorem 6.2.3. Let P be a full-dimensional binary polytope in Rn. Let F be a

facet of P and suppose that v = 〈α1, . . . , αn〉 is an integrally scaled normal vector

for F . Then there exists some isometry A with A : v 7→ 〈β1, . . . , βn〉 such that∑k∈K

βk ∈M(n)

for every collection of indices K ⊆ {1, . . . , n}.

Proof. Suppose by contradiction that there does not exist any isometry that maps

v to a vector 〈β1, . . . , βn〉 satisfying∑

k∈K βk ∈M(n) for all collections of indices

K. Consider an isometry that maps F to a facet that passes through (−1, . . . ,−1).

Let 〈b1, . . . , bn〉 be the image of v under this isometry. We know that we have a

matrix satisfying

det

e1 · · · en

γ21 · · · γ2n...

...

γn1 · · · γnn

= 〈b1, . . . , bn〉

Where γij ∈ {0, 1} for all i, j. However, we know that for some K ⊆ {1, . . . , n} we

have∑k∈K

bk /∈M(n). Replace the top row of the matrix above with 0 − 1 binary

values such that

ei =

{1 if i ∈ K0 otherwise

By construction, this results in a 0−1 binary n×n matrix with determinant∑k∈K

bk.

70


As this value is not in M(n) we have a contradiction. �

This theorem will allow us to further restrict the values of the coefficients of the

polynomials that define the facets of our polytopes. As much of our work will focus

on low-dimensional problems we offer an additional observation for R4.

Corollary 6.2.4. Let P ⊂ R4 be a full-dimensional binary polytope. Let F be a

facet of P . Neither the vector v = 〈2, 2, 1, 1〉 nor any signed permutation thereof

may be a normal vector for F .

Proof. Let v′ be a signed permutation of v and suppose by contradiction that v′

is a normal vector for F . We may apply an isometry that maps F to a facet F ′

passing through (−1,−1,−1,−1). Let 〈β1, . . . , βn〉 be the image of v′ under this

isometry. As before, we have a matrix satisfying

det

e1 e2 e3 e4

γ21 γ22 · · · γ24...

...

γ41 · · · · · · γ44

= 〈β1, β2, β3, β4〉

where γij ∈ {0, 1} for all i, j. As discussed in Theorem 6.2.3, we know that∑k∈K

βk ∈M(4) for all K ⊆ {1, 2, 3, 4}. Notice that M(4) = {0,±1,±2,±3}. Con-

sequently, we may claim that 〈β1, β2, β3, β4〉 must be a (unsigned) permutation of

〈2,−2, 1,−1〉. Notice that the points

(1, 1, 1, 1), (1, 1,−1,−1), (−1,−1, 1, 1), and (−1,−1,−1,−1)

are, following this unsigned permutation, the only binary points in R4 that may

lie on F ′. However, these points do not define a 3 dimensional hyperplane in R4.

This is a contradiction. Hence, no signed permutation of v can be a normal vector

of a facet of P . �

71


As demonstrated above, the knowledge of the distribution of determinant values

for Hadamard matrices is of great use to us. For problems of low to medium dimension

the setsM(n) are readily available. We will refer to the Online Encyclopedia of Inte-

ger Sequences (OEIS) for much of our data regarding the integers in the M(n) sets.

For problems in high dimension it is likely that the exact distribution of Hadamard

determinant values is unknown. In these cases, we remark that the Hadamard bound

holds for matrices of all sizes. As such, the superset

{k ∈ Z | |k| ≤ h where h is the Hadamard bound for n}

may be substituted for M(n) whenever data is unavailable. We now formulate our

necessary condition.

Theorem 6.2.5. Let I ⊆ R[x] be a binary ideal corresponding to a full-dimensional,

proper binary polytope P in Rn. Let `(x) = a1x1 + · · ·+ anxn + a0 be a linear poly-

nomial and suppose that

f(x) = β12x1x2 + · · ·+ β(n−1)nxn−1xn + lower degree terms

is the symbolically-computed normal form for `2(x) in R[x]/I with respect to a

graded monomial order. Note that the βij = βij(a1, . . . , an) terms are functions of

the coefficients of `(x). If I is (1, 1)-SOS then the system

β12(a1, . . . , an) = 0

...

β(n−1)n(a1, . . . , an) = 0∑i 6=j

a2i a2j ≥ 1

a1, . . . , an ∈M(n− 1)

has a solution in Zn.

Proof. Suppose that I is (1, 1)-SOS. Since P is a proper binary polytope we know

72


that some facet F of P is given by a linear hyperplane `(x) = a1x1 + · · ·+ anxn +

a0 that properly depends on at least two variables. Since `(x) must be pseudo-

idempotent in R[x]/I we know that the normal form of `2(x) must be linear.

Consequently, the coefficients of `(x) must satisfy βij(a1, . . . , an) = 0 for all i, j =

1, . . . , n with i 6= j. Moreover, since at least two coefficients of `(x) are non-zero

we know that∑

i 6=j a2i a

2j ≥ 1 must be satisfied as well. Finally, after integrally

scaling the values, we know that each ai may be regarded as the determinant of a

0 − 1 binary (n − 1) × (n − 1) matrix. As a result, we must have ai ∈ M(n − 1)

for all i = 1, . . . , n. It follows that (a1, . . . , an) is a feasible solution to the system

above. �

Some of the constraints in the system from the preceding theorem warrant further

exploration. First, as noted above, we may not have information for the integers in

M(n) for higher values of n. In this case one may resort using the Hadamard bound

for the constraint on our decision variables. Additionally, for an arbitrary binary

ideal it is very likely that Corollary 5.3.11 will apply. In such cases we can replace

the∑

i 6=j a2i a

2j ≥ 1 condition with the stronger constraint∑

i 6=j 6=k 6=i

a2i a2ja

2k ≥ 1

Additionally, we may subject the solutions to the system in Theorem 6.2.5 to the

condition stipulated in Theorem 6.2.3. Constructing a polynomial model of the re-

striction from Theorem 6.2.3 can be cumbersome, but it is an option for problems in

lower dimensions.

There are a handful of strategies that one might employ to determine if a system

such as the one in Theorem 6.2.5 is feasible. We will provide a comparison of a

selection of these strategies in a later chapter. For now we return to Example 6.0.8

to complete the problem.

Example 6.2.6. Again consider the polytope P ⊆ R3 pictured below

In Example 6.0.8 we established that the general affine polynomial `(x) =

73


(1, 1,−1)

(−1, 1,−1)

(1,−1, 1)

(−1, 1, 1)

(−1,−1,−1)

Figure 6.2: An illustration of necessary conditions (cont)

ax1 + bx2 + cx3 + d yields

`2 ≡ (2ab− 2ac+ 2bc)x2x3 + lower degree terms mod I

Following the outline in Theorem 6.2.5 we construct the system

ab− ac+ bc = 0

a2b2 + a2c2 + b2c2 ≥ 1

a, b, c ∈M(2) = {−1, 0, 1}

It is not difficult to see that this system has no solution. Hence, the ideal corre-

sponding to P cannot be (1, 1)-SOS.

It is worth observing that, by Corollary 5.3.11, we could strengthen the system in

the example above by including the constraint a2b2c2 ≥ 1. Though it is not necessary

for this problem, this strengthened condition can be a great benefit in general. We

will discuss algorithmic approaches to recognize the infeasibility of these systems in

the final chapter. We first highlight a collection of specific examples that illuminate

some important concepts regarding the generators of (1, 1)-SOS ideals.

74


Chapter 7

Some Important Counter-Examples

We take a moment to explore a handful of examples that will provide some helpful

insight into the nuances of the facet-generator relationship discussed in Chapter 5.

Most of these will serve as counter-examples to claims that one may be tempted to

make regarding the appearance of (1, 1)-SOS ideals.

7.1 Nuances of Facets and Generators

We will begin with a simple example to illustrate that the presence of bivariate

generators does not guarantee the (1, 1)-SOS property in Rn. In the case that n = 3

it is true that every binary ideal with a generator (in reduced Grobner basis form) that

depends on exactly two variables is (1, 1)-SOS. Up to isometry there is only one such

polytope on 6 vertices, on 5 vertices there are only two polytopes that are isometrically

distinct, and every polytope on 4 variables in R3 is simplicial and hence (1, 1)-SOS by

Theorem 5.2.7. One may exhaustively demonstrate the (1, 1)-SOS property of these

polytopes without much trouble. Though it is probably intuitive that this does not

hold for general n, we provide a counter-example for good measure.

Example 7.1.1. Consider the polytope P ⊂ R4 with vertices given by the collec-

tion of points below

(1, 1,−1, 1), (1, 1,−1,−1), (1,−1, 1,−1), (−1, 1, 1,−1), (1,−1,−1, 1), (−1, 1,−1, 1)

(1,−1,−1,−1), (−1, 1,−1,−1), (−1,−1, 1,−1), (−1,−1,−1, 1), (−1,−1,−1,−1)

Using elimination polynomials we may construct the corresponding ideal I and

compute a reduced Grobner basis for graded lex order with x1 > x2 > x3 > x4.

75


We obtain

I = 〈x21 − 1, x22 − 1, x23 − 1, x24 − 1, x3x4 + x3 + x4 + 1,

x1x2x3 + x1x2 + x1x3 + x2x3 + x1 + x2 + x3 + 1〉

Notice that since I has a generator on three variables we know by Corollary 5.3.11

that P must have a facet that is given by a polynomial that properly depends on

at least three variables. Since x3x4 +x3 +x4 +1 is the only distinguished generator

of I that will be practical in the study of idempotence of linear polynomials, and

this generator only has one bilinear term, we see that we clearly cannot achieve

idempotence of a linear polynomial on three variables in R[x]/I. Consequently, Icannot be (1, 1)-SOS.

For our next example we turn our attention to Corollary 5.3.11. This is the claim

that the presence of a generator on three or more variables indicates the presence

of a facet on three or more variables. As these claims are of such great utility we

are strongly motivated to seek improvements to this statement. We have seen in

Example 5.3.3 that the presence of a distinguished generator that properly depends

on variables xi1, . . . , xik does not ensure the presence of a facet that properly depends

on exactly these variables. We are inspired to ask, “is there at least a facet that

properly depends on a superset of these variables, {xj1, . . . , xjm} ⊇ {xi1, . . . , xik}?”

Unfortunately, though heuristic data suggest that this is very often the case, we

cannot claim that it holds in general. A simple counter-example follows.

Example 7.1.2. Let P ⊆ R3 be the binary polytope pictured below.

Using exclusion polynomials we construct the corresponding ideal I. In reduced

Grobner basis form we have

I = 〈x21−1, x22−1, x23−1, x2x3+x2+x3+1, x1x3+x1+x3+1, x1x2+x1−x2−1〉

One can inspect the figure above to see that, although P has facets that properly

depend on {x1, x2} and {x1, x3}, there is no facet that properly depends on {x2, x3}or any superset thereof.

76


(1,−1,−1)

(−1,−1, 1)

(1, 1,−1)

(−1,−1,−1)

Figure 7.1: Nuances of the facet-generator correspondence

We proceed with another example in the same vein. The correlation between

facets and generators that properly depend on two variables has been established in

Theorem 5.3.1 and the discussion thereafter. Since we typically restrict our focus to

distinguished generators and facets of full-dimensional polytopes we know that we

need not explore univariate polynomials. As such, polynomials that depend on two

variables are minimal, in a manner of speaking. It is natural to ask if polynomials

on the maximal number of variables are equally well-behaved. In other words, we

ask, “if I has a generator that depends on all n variables, does P have a facet that

depends on all n variables?”. Regrettably, the answer to this is no as well. We present

a counter-example below.

Example 7.1.3. Consider the binary ideal in R[x1, x2, x3, x4] given in exclusion

polynomial form below.

I = 〈x21−1, . . . , x24−1, (x1−1)(x2−1)(x3−1)(x4−1), (x1−1)(x2+1)(x3−1)(x4+1),

(x1+1)(x2+1)(x3+1)(x4−1), (x1+1)(x2+1)(x3−1)(x4+1), (x1+1)(x2−1)(x3−1)(x4−1),

(x1 + 1)(x2 + 1)(x3 + 1)(x4 + 1)〉

Computing a reduced Grobner basis for graded lex order we obtain the following

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distinguished generator (amongst others)

x1x3x4 − 2x1x2 − x1x3 − x1x4 + x3x4 − x1 − 2x2 − x3 − x4 − 1

So I contains a generator on all four variables. However, the distinguished facets

for the corresponding polytope are given by

1 + x2 + x3 + x4 ≥ 0, 1− x2 + x3 − x4 ≥ 0,

1− x1 − x2 − x3 ≥ 0, 1− x1 − x2 − x4 ≥ 0

So no facet of the polytope properly depends on all four variables.

In Proposition 5.2.3 we observed that a generator with a bilinear leading term

is necessary for a proper binary ideal to achieve the (1, 1)-SOS property. Much of

our work in Chapter 5 served to demonstrate how the arrangement of the bilinear

terms in these quadratic generators is critical to our analysis. Indeed, generators of

degree 3 or greater will be of no use in the study of idempotence of linear polynomials

in the quotient ring for a binary ideal. One may naturally wonder if the presence

of such unwieldy generators inherently disrupts the (1, 1)-SOS structure of a general

binary ideal. We ask, “can the reduced Grobner basis for a (1, 1)-SOS ideal contain

generators of degree three or greater?” The example below shows that the answer is

yes.

Example 7.1.4. Consider the binary polytope P ⊆ R4 with corresponding ideal

given below in exclusion polynomial form.

I = 〈x21−1, . . . , x24−1, (x1+1)(x2+1)(x3+1)(x4−1), (x1+1)(x2+1)(x3−1)(x4+1),

(x1+1)(x2−1)(x3+1)(x4+1), (x1−1)(x2+1)(x3+1)(x4+1), (x1+1)(x2−1)(x3−1)(x4−1),

(x1−1)(x2+1)(x3−1)(x4−1), (x1−1)(x2−1)(x3+1)(x4−1), (x1−1)(x2−1)(x3−1)(x4+1)〉

Computing a reduced Grobner basis for graded lex order we obtain

I = 〈x21 − 1, . . . , x24 − 1, x1x4 − x2x3, x1x3 − x2x4, x1x2 − x3x4, x2x3x4 − x1〉

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We emphasize that the reduced Grobner basis for I contains a generator of degree

three. If we compute the distinguished facets of P we have

x1 + x2 + x3 − x4 + 2 ≥ 0, x1 + x2 − x3 + x4 + 2 ≥ 0, −x1 + x2 + x3 + x4 + 2 ≥ 0

−x1−x2−x3 +x4 + 2 ≥ 0, −x1 +x2−x3−x4 + 2 ≥ 0, x1−x2 +x3 +x4 + 2 ≥ 0

−x1 − x2 + x3 − x4 + 2 ≥ 0, x1 − x2 − x3 − x4 + 2 ≥ 0

Let us consider the facet given by `(x) = x1− x2− x3− x4 + 2 ≥ 0. If we compute

the normal form of the square of this polynomial we obtain

(x1 − x2 − x3 − x4 + 2)2 ≡ 4x1 − 4x2 − 4x3 − 4x4 + 8 mod I

Hence, `(x) is pseudo-idempotent in the quotient ring R[x]/I. One may proceed

to show that this holds for the remaining distinguished facets of P . Hence, P is

(1, 1)-SOS by Corollary 4.2.2.

7.2 Projections of Polytopes

Next we take a moment to explore the behavior of the polytopes corresponding to

(1, 1)-SOS ideals under projection. Given a set S ⊆ Rn let us denote the projection

off of the ith coordinate as follows

πi(S) = {(x1, . . . , xi−1, xi+1, . . . , xn) ∈ Rn−1 | (x1, . . . , xi−1, xi, xi+1, . . . , xn) ∈ S}

The geometric notion of the projection of a variety is closely related to the algebraic

notion of elimination ideals. We remind the reader that, given an ideal I ⊆ R[x], the

ith elimination ideal of I is

Ii = I ∩ R[x1, . . . , xi−1, xi+1, . . . , xn]

That is, Ii consists of the polynomials in I that do not properly depend on xi. The

elimination ideal of a given ideal I can be computed using Grobner bases with re-

spect to elimination orders (pure lexicographic order is commonly used). Let G be

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a Grobner basis for I with respect to pure lex order with x1 > · · · > xn. Then

G ∩ R[x2, . . . , xn] is a Grobner basis for I1. We refer to [1] for proof.

Suppose that V ⊆ Rn is the variety of an ideal I ⊆ R[x]. It is natural to ask if

the relationship between I and V extends to a relationship between Ii and πi(V ).

We can make use of the containment

πi(V ) ⊆ V(Ii)

Equality does not hold for these sets in general since πi(V ) may not be a variety [7].

We can, however, state that V(Ii) is the Zariski closure of πi(V ), i.e. V(Ii) is the small-

est variety containing πi(V ). However, if we restrict our focus to zero-dimensional

ideals (in particular, binary ideals) we can be sure that the containment above holds

at equality since every finite set of points is a variety.

Projection techniques are frequently employed in the analysis of polytopes [23].

For the case of binary polytopes we will show that we cannot claim consistency of the

(1, 1)-SOS property (or lack thereof) of the corresponding ideals under projection.

First, we observe that every binary polytope in R2 is (1, 1)-SOS. Hence, it is possible

(and, indeed, common) for a polytope that is not (1, 1)-SOS to project onto a (1, 1)-

SOS polytope of lower dimension. The reverse claim is more interesting. Heuristics

show that if Ii is not (1, 1)-SOS for some i then the original ideal I very frequently

fails to be (1, 1)-SOS as well. However, counter-examples do exist. One is provided

below.

Example 7.2.1. Let P ⊆ R4 be the polytope with the following vertices

(1, 1, 1,−1), (1, 1,−1, 1), (1,−1, 1, 1), (−1, 1, 1,−1)

(−1, 1,−1, 1), (1,−1,−1,−1)

We compute the corresponding ideal and, for graded lex order, we have

I = 〈x21 − 1, . . . , x24 − 1, x3x4 + x2, x2x4 + x3,

x2x3 + x4, x1x3 + x1x4 − x3 − x4, x1x2 − x1 − x2 + 1〉

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The distinguished facets of P can be shown to be

1− x2 − x3 − x4 ≥ 0, x1 + x2 ≥ 0, 1 + x2 + x3 − x4 ≥ 0

1− x2 + x3 + x4 ≥ 0, 1 + x2 − x3 + x4 ≥ 0

Each of the linear polynomials above is pseudo-idempotent in R[x]/I. Hence, I is

(1, 1)-SOS. Consider projecting P off of the x2 coordinate. We compute a Grobner

basis for I with respect to lex order with x2 > x1 > x3 > x4 and we discard any

generator in which x2 appears. We obtain

I2 = 〈x21 − 1, x23 − 1, x24 − 1, x1x3 + x1x4 − x3 − x4〉

Note that the fact that this matches the elimination of the x2 variable in the

graded lex basis for I is coincidental. Notice that the polytope corresponding to

I2 is isometric to the polytope from Example 5.3.12. By Corollary 5.3.11 we may

claim that I2 cannot be (1, 1)-SOS. So, although uncommon, it is possible for a

(1, 1)-SOS polytope to project onto a lower-dimensional polytope that fails to be

(1, 1)-SOS.

81


Chapter 8

Comparison of Techniques

In this chapter we will provide a comparison of the algebraic approach to the recogni-

tion of the (1, 1)-SOS property as described in chapters 4, 5, and 6 and the geometric

techniques discussed in chapter 3. We will be interested in both the CPU running

time and the percentage of inconclusive results for our algebraic analysis. The per-

formance of the geometric approach is very sensitive to the original presentation of

the problem. Both the case in which we are given a set of generators of an ideal and

seek to determine whether or not the ideal is (1, 1)-SOS and the case in which we are

given the set of vertices of a polytope and seek to determine the (1, 1)-SOS status of

the corresponding ideal will be explored. Before analyzing these results we discuss

some formatting strategies.

8.1 Computational Strategies

We aim to compile a code that will allow us to quickly and effectively apply the

sufficient conditions and necessary conditions provided by our algebraic analysis to

problems involving the SOS structure of ideals. Many of these results, such as Propo-

sition 5.3.13 on compliance of bivariate facets and the results based upon standard

monomials such as Theorem 5.2.6, can be implemented without much difficulty. Oth-

ers, particularly the necessary condition presented in Theorem 6.2.5, warrant some

discussion.

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We have established that an ideal I ⊆ R[x] is (1, 1)-SOS only if the system

β12(a1, . . . , an) = 0

...

β(n−1)n(a1, . . . , an) = 0∑i 6=j

a2i a2j ≥ 1

a1, . . . , an ∈M(n− 1)

(8.1)

has a solution in Zn where the indeterminants ai are coefficients of a generic linear

polynomial `(x) = a1x1+· · ·+anxn+a0 and the polynomials βij are the coefficients of

the bilinear monomials xixj in the normal form of `2(x) in R[x]/I. Notice that, due

to the inequality, the system in 8.1 describes a semi-algebraic set and not a variety.

If we wish to apply many of our tools from algebraic geometry it is helpful to rewrite

the system as follows:

β12(a1, . . . , an) = 0

...

β(n−1)n(a1, . . . , an) = 0(∑i 6=j

a2i a2j

)− t2 − 1 = 0

a1, . . . , an ∈M(n− 1)

(8.2)

Notice that formulation 8.1 has a solution if and only if formulation 8.2 has a solution

with t ∈ R. Moreover, the constraints ai ∈ M(n − 1) can be quickly encoded as

polynomial equations of the form ∏m∈M(n−1)

(ai −m) = 0

In this manner, our question about the feasibility of system 8.1 may be regarded

as a question about the real variety of the ideal formed by the polynomials in 8.2.

For a given ideal I let us denote by I∗ ⊆ R[a1, . . . , an, t] the ideal generated by the

polynomials in the corresponding system 8.2. Then I is (1, 1)-SOS only if VR(I∗) 6= ∅.

83


By the Real Nullstellensatz, it follows that I is (1, 1)-SOS only if the real radical idealR√I∗ is not the entire ring R[a1, . . . , an, t]. As such, one strategy for determining the

feasibility of 8.1 is to check if 1 ∈ R√I∗. However, as discussed in chapter 2 the real

radical algorithm can be unpredictable and expensive for many ideals. Our heuristics

suggest that it is often best to avoid the computation of real radical ideals when other

methods are available. For our purposes we can streamline this computation using a

classic theorem regarding zero-dimensional ideals.

Theorem 8.1.1. Let I ⊆ C[x1, . . . , xn] be an ideal. Then V(I) is finite if and

only if I ∩ C[xi] 6= 〈0〉 for all i = 1, . . . , n.

Proof. Suppose that V(I) = {p1, . . . , pk}. For i ∈ {1, . . . , n} and j ∈ {1, . . . , k}let pij denote the ith coordinate of pj. For each i we construct the polynomial

fi(xi) =k∏j=1

(xi − pij)

Then fi(xi) is a univariate polynomial that vanishes on all points in V(I). Hence,

fi(xi) ∈√I meaning fmi (xi) ∈ I for some m ∈ N.

Now suppose that I contains a univariate polynomial fi(xi) for all i = 1, . . . , n.

Let Pi denote the set of roots of fi(xi). Then the set

P = {(p1, . . . , pn) ∈ Cn | pi ∈ Pi for all i}

is the variety of the ideal 〈f1, . . . , fn〉. Since 〈f1, . . . , fn〉 ⊆ I we know that V(I) ⊆P . As P is finite, V(I) must be finite as well. �

Regarding the system 8.2 we note that the constraints ai ∈M(n− 1) ensure that

each ai can take on only finitely many values. Hence, the relationship t2 =∑i 6=j

a2i a2j − 1

ensures that t can take on only finitely many values as well. Thus, I∗ must be

zero-dimensional. We may compute the minimal univariate polynomial for t using

elimination ideals. A Grobner basis for I∗ with respect to pure lexicographic order

with a1 > · · · > an > t will contain a univariate polynomial in t as a generator. Let

f(t) ∈ I∗ denote this univariate polynomial. We conclude that I is (1, 1)-SOS only if

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f(t) has real roots. Our experience shows that testing f(t) for real roots is preferable

to a complete computation of R√I∗. We will employ this strategy in our code for the

algebraic approach to the recognition of (1, 1)-SOS ideals.

8.2 Data and Analysis

In this section we present the data collected in the comparison of algebraic and ge-

ometric approaches to the analysis of SOS structure of randomly generated binary

ideals. The algebraic approach consists of a sequence of tests to determine whether

or not the ideal in question satisfies any of the sufficient conditions or violates any of

the necessary conditions presented in the preceding chapters. The specific codes used

are presented in the following section for reference. The geometric approach consists

of computing the facets of the corresponding polytope and testing the facet-defining

polynomials for idempotence in the quotient ring for the given ideal. Again, the codes

utilized are included in the final section of this chapter.

We have collected data for the CPU time necessary to recognize SOS structure for

the algebraic techniques, denoted Algebraic, and we have subdivided the tasks for the

geometric approach to illuminate some specific sensitivities. Given an ideal, the geo-

metric approach to SOS analysis requires us to compute the corresponding polytope.

This requires the computation of the real variety of the ideal. We denote this task by

Variety. Notice that this computation is considerably more expensive than many of

the other steps in the analysis. As such, it is of interest to distinguish between SOS

problems that initially present an ideal and problems that initially present a polytope

(given as the defining set of vertices) as this expensive computation is not necessary

in the latter case. The geometric approach then requires us to compute the facets of

the polytope corresponding to our ideal. This process is known as facet enumeration

and, for polytopes in Rn on m vertices, operates on the order O(mn2 ) whenever n ≥ 4

is fixed [6]. We denote this task by Enumeration and we will use the software package

LRS [21] to perform these computations. Finally, the geometric approach requires

us to test the facet-defining inequalities for idempotence in the quotient ring for the

given (or constructed) ideal. This step is denoted Idempotence.

When studying problems in which we are given a binary ideal and asked to de-

85


termine the SOS status we do not assume that the ideal is in Grobner basis form.

Hence, the Grobner basis computation is included in the running time presented be-

low. Also, it is of interest to study the behavior of these techniques on polytopes of

varying sizes. Hence, rather than simply organizing our data according to the dimen-

sion of these polytopes, we subdivide the results according to the number of vertices

in these polytopes. We will see that the algebraic approach performs particularly well

in the analysis of ideals corresponding to relatively large polytopes.

We first present the data collected for ideals and polytopes in lower dimensions.

Table 8.1: Average Running Times for Low Dimensions (seconds)(Dimension, |vert(P)|) Algebraic Variety Enumeration Idempotence

(3,7) 0.0257 0.1710 0.0280 0.0357(3,6) 0.0827 0.1347 0.0350 0.0360(3,5) 0.0930 0.0930 0.0306 0.0460(3,4) 0.0360 0.1037 0.0261 0.0457(4,15) 0.0257 0.2397 0.0280 0.0460(4,14) 0.0257 0.2647 0.0347 0.0460(4,13) 0.0460 0.2593 0.0873 0.0513(4,12) 0.0467 0.2750 0.0373 0.0460(4,11) 0.0883 0.2543 0.0369 0.0460(4,10) 0.1350 0.2540 0.0390 0.0513(4,9) 0.1450 0.2543 0.0371 0.0673(4,8) 0.1450 0.2387 0.0436 0.0673(4,7) 0.1713 0.2180 0.0371 0.0827(4,6) 0.4400 0.2283 0.0330 0.0777(4,5) 0.0673 0.2180 0.0330 0.0727(5,31) 0.0567 0.5143 0.0369 0.0513(5,26) 0.0777 0.4627 0.0459 0.0880(5,24) 0.0780 0.4937 0.0366 0.1193(5,22) 0.0727 0.4730 0.0501 0.0880(5,17) 0.1347 0.4363 0.0459 0.1557(5,16) 0.1453 0.4467 0.0414 0.1610(5,14) 0.2750 0.4520 0.0501 0.1817(5,10) 0.6067 0.4623 0.0479 0.2597(5,6) 0.2647 0.4830 0.0281 0.2910

86


Before we proceed with additional data we highlight a handful of interesting results

from the collection above. First, we note that there is a considerable spike in the

running time for the algebraic techniques in the cases (4, 6) and (5, 10). This can

be explained when we turn our attention to measuring the accuracy of the algebraic

method. The geometric approach to SOS analysis has a considerable advantage in

that the (1, 1)-SOS status of a given ideal can always be determined without any

risk of inconclusive results. Our algebraic approach comprises a series of sufficient

conditions and necessary conditions and, consequently, it is possible for certain ideals

to navigate these tests without violating any necessary conditions nor satisfying any

sufficient conditions. Our heuristics have shown that the risk of inconclusive results

for the algebraic tests is substantially reduced when we restrict our focus to binary

polytopes in Rn that occupy a relatively large percentage of the ambient hypercube.

Let us use the following definition to indicate when a polytope has more than half of

the 2n available binary vertices in Rn.

Definition 8.2.1. Let P ⊂ Rn be a full-dimensional binary polytope. We will say

that P is large if |vert(P)| > 2n−1. We will call P small otherwise.

The failure rate for the algebraic approach across all of our computations was an

unrealistically high 14.6%. However, if we restrict our focus to large polytopes the

percentage of inconclusive results improves to 1.8%. Additionally, the vast majority of

inconclusive results occurred in polytopes with roughly 2n vertices, with percentages

leaping above 50% in the case of (4, 6) and (5, 10). The spike in computation times

in the table above is the result of the large percentage of ideals that surmounted the

entirety of the algebraic tests without producing a conclusive result. In light of this

behavior we are motivated to focus on ideals corresponding to large polytopes as we

proceed to higher dimensions.

As one can see, the gap between the running times for the algebraic and geometric

methods of analysis becomes more pronounced in R6 and R7. Additionally, the per-

centage of inconclusive results among large polytopes steadily improved from 4.1% in

R5 to 2.8% in R6, and we encountered no inconclusive results at all across all of the

87


Table 8.2: Average Running Times for Large Polytopes in R6 and R7 (seconds)(Dimension, |vert(P)|) Algebraic Variety Enumeration Idempotence

(6,54) 0.1507 1.3723 0.1061 0.1867(6,49) 0.2750 1.6170 0.0970 0.3740(6,44) 0.3377 1.4350 0.0969 0.4207(6,39) 0.5457 1.2320 0.0921 0.5453(6,34) 0.6757 1.2890 0.0747 0.7797(7,115) 0.4367 8.8500 1.5477 0.5193(7,105) 0.8057 10.0617 1.3767 0.9150(7,95) 1.7260 9.4013 1.1498 1.8353(7,85) 2.4437 8.4757 0.9952 2.7870(7,75) 3.7387 8.4393 0.7882 3.9307(7,65) 5.6750 9.1517 0.6147 6.5047

randomly generated large polytopes in R7.

Our overall results are summarized according to dimension in the following two

tables. We have distinguished between the running times for the geometric method in

the case that the problem is initially presented with a polytope, denoted Geom. given

Polytope, and initially presented with an ideal, denoted Geom. given Ideal. Notice

that the necessary computations for the algebraic approach are not influenced by this

distinction.

Table 8.3: Summary of Average Running Times for Low Dimension (seconds)Dimension Algebraic Geom. given Polytope Geom. given Ideal

3 0.0593 0.0708 0.19634 0.1215 0.1016 0.34755 0.1901 0.1976 0.6669

The following table summarizes our results by dimension when we restrict our

attention to large polytopes.

In conclusion, an algebraic approach to SOS analysis can be performed very

quickly, especially when we seek to determine if a given ideal is (1, 1)-SOS. When

88


Table 8.4: Summary of Average Running Times for Large Polytopes (seconds)Dimension Algebraic Geom. given Polytope Geom. given Ideal

3 0.0671 0.0704 0.20334 0.0732 0.0943 0.35225 0.0839 0.1435 0.61956 0.3969 0.5546 1.94377 2.4709 3.8274 12.8907

we seek to determine if a given polytope corresponds to a (1, 1)-SOS ideal the alge-

braic approach still performs competitively with the geometric techniques, although

the difference is not quite as pronounced. The significant drawback to the algebraic

techniques lies in the risk of inconclusive results. In situations in which a researcher

suspects or knows that his or her polytope contains a relatively large number of ver-

tices the risk of failure is likely to be low and the algebraic method may be favorable,

especially when initially presented with an ideal.

8.3 Codes

In this section we present the codes used to collect the data that were presented in

the preceding section. We have chosen Maple as the primary vehicle for our analysis

since the tools of algebraic geometry are implemented in a convenient manner therein.

We have assumed throughout that the Groebner, PolynomialIdeals, Combinat, and

RootFinding packages have been loaded. We will also utilize tools in the LinearAlge-

bra package but these will be called locally as this package causes conflicts with the

Groebner package.

Our instances will be crafted from randomly generated lists of binary points in Rn.

To facilitate the construction of our ideals via exclusion polynomials we will interpret

the binary points to be those that are not present in the polytope. So the input

P := [[1, 1, 1], [−1,−1,−1]]

will yield the polytope in R3 on the six vertices that are not present in P .

Our first code is a procedure titled MakeIdeal. This will be used to convert the

89


list of points P into the corresponding ideal.

MakeIdeal := proc(P) local i, f, j, J, ord, G;

for i from 1 to nops(P) do

f[i]:= 1;

end do;

for j from 1 to nops(P) do

for i from 1 to nops(P[1]) do

f[j]:= f[j]· (x[i] + P[j][i]);

end do;

end do;

J:=〈0〉;

for j from 1 to nops(P[1]) do

J:= Add(J,x[j]2 - 1);

end do;

for i from 1 to nops(P) do

J:= Add(J,f[i]);

end do;

ord:=x[1];

for i from 2 to nops(P[1]) do

ord:=ord,x[i];

end do;

G:=Basis(J,grlex(ord));

return G;

end proc:

Notice that the MakeIdeal code will return the Grobner basis for the ideal corre-

90


sponding to the polytope given by P . For some of our subsequent codes it will be

necessary to bear in mind that the set G is distinct from the ideal generated by G.

Next we have a code that will ensure that the polytope given by P is full-

dimensional. Since G in the output of the preceding code has been generated with

respect to a graded order, this is a simple matter of checking G for linear polynomials.

CheckDim:=proc(G) local i, d, degs, mindeg, result;

for i from 1 to nops(G) do

d[i] := degree(G[i]);

end do;

degs := (2);


degs := degs, d[i];

end do;

mindeg := min(degs);

if mindeg = 1 then

result := NotFullDim;

else result := 0;

end if;

return result;

end proc:

The result = 0 value will be utilized throughout the coding of the sufficient and

necessary conditions to indicate that no terminal conclusion has been reached. Our

next code is a short tool to improve Maple’s coeff command. Computing the coeffi-

cient of a given monomial in a univariate polynomial is a simple task using Maple.

To compute the coefficient of a multivariate (in particular, a bilinear) monomial we

must extend this functionality with a small code.

91


cof := proc(f,m) local v, a, i;

v := indents(m, name); a := f;

for i in v do

a := coeff(a, i, degree(m, i));

end do;

return a;

end proc:

This code will allow us to quickly collect the bilinear terms that appear in the

distinguished quadratic generators of our ideal. Recall that these are the terms that

are instrumental in the (1, 1)-SOS ideals. The following code counts these bilinear

terms. Notice that the dimension n of the ambient space is taken as input here.

CountBiLin := proc(G, n) local count, i, j, k;

count := 0;


if degree(G[i]) < 3 then

for j from 1 to n-1 do

for k from j+1 to n do

if cof(G[i],x[j]·x[k]) = 0 then

count := count + 0;

else

count := count + 1;

end if;

end do;

end do;

end if;

end do;

return count;

end proc:

92


Next we construct a code that will return the maximum number of variables

present in a single generator of our ideal. This can be employed to anticipate suf-

ficient conditions discussed in the facet-generator correspondence section of chapter 5.

MaxVars := proc(G) local vars, i, numvar, maxvars;

vars := (1);


numvar[i] := nops(indets(G[i]));

end do;


vars := vars, numvar[i];

end do;

maxvars := max(vars);

return maxvars;

end proc:

Our next code serves as an implementation of the necessary condition described in

Theorem 6.2.5. Notice that the values forM(n) here can be easily extended to meet

the researchers needs using either the Hadamard bound or, more specifically, the dis-

tribution of positive values for determinants of binary matrices whenever information

is available. Additionally, this code has been tailored to the results of Corollary 5.3.11

which may be verified using the MaxVars code above. This code has been written

under the assumption that the necessary condition via IP feasibility, being the most

expensive to check, will be applied last.

CheckIP := proc(G,n) local ell, i, ord, norm, j, c, f, k, IP, M, g, result, B;

ell := a[n+1];

for i from 1 to n do

93


ell := ell + a[i]·x[i];

end do;

ord := x[1];


ord := ord, x[i];

end do;

norm := NormalForm(ell2, 〈G〉, grlex(ord));

for i from 1 to n-1 do

for j from i+1 to n do

c[i][j] := coeff(coeff(norm, x[j]), x[i]);

end do;

end do;

f := 0;


for j from 2 to n-1 do

for k from 3 to n do

f := f + a[i]2·a[j]2·a[k]2;

end do;

end do;

end do;

if n = 3 then M := [1] end if;

if n = 4 then M := [1,2] end if;

if n = 5 then M := [1,2,3] end if;

if n = 6 then M := [1,2,3,4,5] end if;

if n = 7 then M := [1,2,3,4,5,6,7,8,9] end if;


g[i] := a[i];

end do;

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for j in M do

g[i] := g[i]·(a[i]2 - j2);

end do;

end do;

IP := (f - t2 - 1);


IP := IP, g[i];

end do;


for j from i+1 to n do

IP := IP, c[i][j];

end do;

end do;

ord := a[1];


ord := ord, a[i];

end do;

ord := ord, t;

B := Basis( 〈IP〉, plex(ord));

for i from 1 to nops(B) do

if indets(B[i]) = {t} then f := B[i];

end if;

end do;

if HasRealRoots(f) then

result := Inconclusive

else result := NotSOS

end if;

return result;

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end proc:

With these tools we may construct a code that will subject a given ideal (or poly-

tope) to a series of sufficient conditions and necessary conditions that aim to identify

the (1, 1)-SOS property or lack thereof. The code follows.

AlgTest := proc(P) local G, result, n, count, maxvars, IP, degcount, i;

G := MakeIdeal(P);

result := CheckDim(G);

n := nops(P[1]);

if result = 0

then count := CountBiLin(G,n);

if count = 0 then result := NotSOS;

end if;

end if;

if result = 0

then maxvars := MaxVars(G);

if maxvars = 2 then result := SOS;

end if;

end if;

if result = 0 then

degcount := 0;

for i from 1 to nops(G) do ;

if degree(G[i]) = 2 then degcount := degcount + 1;

end if;

end do;

if degcount = n + numbcomb(n, 2) then result := SOS;

end if;

end if;

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if result = 0

then result := CheckIP(G, n);

end if;

return result;

end proc:

Our next code is used to measure the time taken to compute the variety of a given

ideal.

FindVariety := proc(G, n) local vars, i eqn, solns;


vars := vars, x[i];

end do;

eqn := G[1] = 0;


eqn := eqn, G[i] = 0;

end do;

solns := solve([eqn], [vars]);

return solns;

end proc:

Finally, we may test the facets of a given polytope for idempotence in the quotient

ring for a given ideal using the following code. Recall that the facet enumeration step

of the geometric approach to our problems is computed using the LRS software [21].

This is an exceptionally popular and optimized package of algorithms for facet enu-

meration, vertex enumeration, convex hull computations, and other techniques used

in the analysis of polytopes. For our purposes, the output produced by LRS for facet

enumeration comes in the form of a matrix A such that Ax ≥ 0 gives a minimal

half-space representation of the polytope in question. Notice here that the vector of

variables x is extended to include the constant 1 in the first entry. We will call this

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matrix M in the following code.

IdempotenceTest := proc(M,P) local dim, G, ord, i, vars, eqns, numfacets, d, result;

G := MakeIdeal(P);

dim := LinearAlgebra[Columndimension](M) - 1;

ord := x[1];

for i form 2 to dim do

ord := ord, x[i];

end do;

vars := Matrix([[1]]);

for i from 1 to dim do

vars := Matrix([[vars], [x[i]]]);

end do;

eqns := LinearAlgebra[MatrixMatrixMultiply](M, vars);

numfacets := LinearAlgebra[RowDimension](M);

for i from 1 to numfacets do

d[i] := degree(NormalForm((eqns[i][1])2, 〈G〉, grlex(ord)));

end do;

result := SOS;

for i from 1 to numfacets do

if d[i] > 1 then result := NotSOS

end if;

end do;

return result;

end proc:

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Bibliography

[1] Adams, William, and Philippe Loustaunau. An Introduction to Grobner Bases.Providence, RI: American Mathematical Society. 1994.

[2] Basu, Saugata, Richard Pollack, and M. Roy. Algorithms in Real Algebraic Ge-ometry. Berlin: Springer, 2003.

[3] Blekherman, Grigoriy, Pablo Parrilo, and Rekha Thomas. Semidefinite Optimiza-tion and Convex Algebraic Geometry. Philadelphia, PA: Society for Industrialand Applied Mathematics. 2012.

[4] Brenner, Joel. “The Hadamard Maximum Determinant Problem.” The AmericanMathematical Monthly 79.6 (1972): 626.

[5] Buchberger, Bruno. “A Criterion for Detecting Unnecessary Reductions in theConstruction of Grobner Bases.” Symbolic and Algebraic Computation LectureNotes in Computer Science (1979): 3 - 21.

[6] Chazelle, Bernard. “An Optimal Convex Hull Algorithm in any Fixed Dimen-sion.” Discrete & Computational Geometry 10.4 (1993): 377 - 409.

[7] Cox, David, Donal O’Shea, and John Little. Ideals, Varieties, and Algorithms.New York: Springer, 1997.

[8] Cox, David, John Little, and Donal O’Shea. Using Algebraic Geometry. NewYork: Springer, 1998.

[9] Conti, Pasqualina, and Carlo Traverso. “Buchberger Algorithm and Integer Pro-gramming”. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,Lecture Notes in Computer Science (1991): 130 - 139.

99


[10] Gouveia, Joao, Pablo Parrilo, and Rekha Thomas. “Theta Bodies of PolynomialIdeals”. SIAM Journal on Optimization 20.4 (2010): 2097 - 118.

[11] Greuel, G., and Gerhard Pfister. A Singular Introduction to Commutative Alge-bra Berlin: Springer, 2002.

[12] Hadamard, J. “Resolution d’une question relative aux determinants”. Bulletindes sciences Mathematiques 17 (1893): 240 - 246.

[13] Lasserre, J. “A Lagrangian Relaxation View of Linear and Semidefinite Hierar-chies”. SIAM Journal on Optimization 23.3 (2013) 1742 - 1756.

[14] Lasserre, J. “Semidefinite Programming vs LP Relaxations for Polynomial Pro-gramming”. Mathematics of Operations Research 27.2 (2002): 347 - 360.

[15] Lasserre, Jean, Kim-Chuan Toh, and Shouguang Yang. “A Bounded Degree SOSHierarchy for Polynomial Optimization”. Journal on Computational Optimiza-tion 1 (2015).

[16] Lasserre, Jean, Monique Laurent, and Philipp Rostalski. “Semidefinite Charac-terization and Computation of Real Radical Ideals”. Foundations of Computa-tional Mathematics 8.5 (2007): 607 - 47.

[17] Lasserre, Jean, Monique Laurent, Bernard Mourrain, et all. “Moment Matrices,Border Bases and Real Radical Computation”. Journal of Symbolic Computation51 (2013): 63 - 85.

[18] Laurent, Monique. “Sums of Squares, Moment Matrices, and Optimization OverPolynomials”. Emerging Applications of Algebraic Geometry The IMA Volumesin Mathematics (2008): 157 - 270.

[19] Lewis, D. J., D. K. Faddeev, I.S. Sominskii, and J. L. Brenner. “Problems inHigher Algebra”. The American Mathematical Monthly 73.5 (1966): 559.

[20] Lovasz, Lazlo. “Semidefinite Programs and Combinatorial Optimization”. RecentAdvances in Algorithms and Combinatorics New York: Springer. 11 (2003) 137- 194.

100


[21] LRS Version 6.1, http://cgm.cs.mcgill.ca/ avis/C/lrslib/USERGUIDE.html

[22] Metropolis, N. “Spectra of determinant values in (0,1) matrices”. Computers inNumber Theory: Proceedings of the Science Research Atlas Symposium London:Academic Press. (1971): 271-276.

[23] Nemhauser, George, and Laurence Wolsey. Integer and Combinatorial Optimiza-tion. New York: Wiley, 1988.

[24] Orrick, William. “The Maximal {−1, 1}-determinant of Order 15”. Metrika 62.2-3 (2005): 195 - 219.

[25] Parrilo, Pablo. Structured Semidefinite Programs and Semialgebraic GeometryMethods in Robustness and Optimization. PhD Thesis, California Institute ofTechnology, May 2000.

[26] Prajna, S., A. Papachristodoulou, and P. Parrilo. “Introducing SOStools: a gen-eral purpose sum of squares programming solver”. Proceedings of the 41st IEEEConference on Decision and Control 1 (2002): 741 - 746.

[27] Putinar, M. “Positive Polynomials on Compact Semi-algebraic Sets”. IndianaUniversity Mathematics Journal 42.3 (1993) 969 - 984.

[28] Schrijver, A. Theory of Linear and Integer Programming. Chichester: Wiley,1986.

[29] Schrijver, A. Combinatorial Optimization: Polyhedra and Efficiency. Berlin:Springer, 2003.

[30] SeDuMi Version 1.3, http://sedumi.ie.lehigh.edu/

[31] Sherali, Hanif, and Warren Adams. “A Hierarchy of Relaxations Between theContinuous and Convex Hull Representations for Zero-One Programming Prob-lems”. SIAM Journal on Discrete Mathematics 3.3 (1990): 411-30.

[32] Sturmfels, Bernd. Solving Systems of Polynomial Equations. Providence, RI:American Mathematical Society, 2002.

101


[33] Stengle, G. “A Nullstellensatz and a Positivstellensatz in Semialgebraic Geome-try”. Math Ann. 207.2 (1974): 87 - 97.

[34] Vinzant, Cynthia. Real Algebraic Geometry in Convex Optimization. PhD Thesis,University of California, Berkeley, Spring 2011.

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