Triangular Sets Seminar

8/2/2019 Triangular Sets Seminar

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Triangular Sets Seminar

Notes by David Wilson


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Contents

I Semester 1 20112012 4

1 Seminar 1: J. H. DavenportOctober 3rd 2011 51.1 What is triangular? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Seminar 2: J. H. DavenportOctober 10th 2011 72.1 Skype Conversation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Other Important People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 What is R-Maple? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Seminar 3: G. K. Sankaran

October 17th 2011 93.1 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 What about maps? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Seminar 4: J. H. DavenportOctober 24th 2011 124.1 Grobner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Triangular Sets and Regular Chains . . . . . . . . . . . . . . . . . . . . . . 134.3 Multiplicity and Positive Dimensions . . . . . . . . . . . . . . . . . . . . . . 14

5 Seminar 5: J. H. Davenport

October 31st 2011 155.1 Degree Reduction Under Specialization . . . . . . . . . . . . . . . . . . . 16

6 Seminar 6: J. H. DavenportNovember 7th 2011 196.1 Degree Reduction under Specialization . . . . . . . . . . . . . . . . . . . 206.2 Grobner Bases under Specializations . . . . . . . . . . . . . . . . . . . . . . 206.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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7 Seminar 7: N. VorobjovNovember 21st 2011 237.1 Semi-monotone sets, monotone functions and maps . . . . . . . . . . . . . . 23

8 Seminar 8: J. H. DavenportNovember 28th 2011 308.1 Admin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2 Computing CAD via TD [CMXY09] . . . . . . . . . . . . . . . . . . . . . . 308.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.4 Another view of CADs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Seminar 9: J. H. Davenport

December 5th 2011 339.1 Last Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2 Continuation of discussion on [CMXY09] . . . . . . . . . . . . . . . . . . . 339.3 Their method for making a CAD . . . . . . . . . . . . . . . . . . . . . . . . 34

9.3.1 InitialPartition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.3.2 MakeCylindrical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.3.3 MakeSemiAlgebraic . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9.4 Example - the parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10 Seminar 10: A. LocatelliDecember 12th 2011 36

10.1 Original Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.2 Alternative View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.3 Chen et als paper [CMXY09] . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.4 F-invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

II Semester 2 20112012 39

11 Seminar 11: J. H. DavenportFebruary 10th 2012 40

12 Seminar 12: D. J. WilsonFebruary 17th 2012 4312.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13 Seminar 13: Informal DiscussionFebruary 24th 2012 45

14 Seminar 14: Informal DiscussionMarch 2nd 2012 46

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15 Seminar 15: James H. DavenportMarch 9th 2012 47

16 Seminar 16: Christopher W. BrownMarch 22nd 2012 50

17 Seminar 17: James H. DavenportMarch 29th 2012 52

18 Seminar 18: James H. DavenportApril 26th 2012 53

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Part I

Semester 1 20112012

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Chapter 1

Seminar 1: J. H. Davenport

October 3rd 2011

The seminar will be loosely based around Triangular Decomposition and the EPSRC grant.The timing will also allow Skype contact with Canadians, and will finish at 4pm to allowtime for JHD to get to 8W to teach.

1.1 What is triangular?

When discussing linear equations, triangular sets refer to upper triangular matrices:

0 0 0 0 0 0 0 0 0

Definition 1.1. A set of polynomials S in k[x1, . . . , xn] is triangular if different elementsof S have different mvars.

Example 1.1. Consider the following two equations:

x2

1

(x + 1)(y 1) + (x 1)(y2 1)

Then this is triangular under the assumption that y > x. However there are only 3solutions, not 4 as you would expect! When x = 1 the second equation becomes linear,whereas when x = 1 the second equation is quadratic.

This is not equiprojectable for x.

This idea is formalized through the idea of regular chains.

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Definition 1.2. A set of polynomials S = {s1, . . . , sk} in k[x1, . . . , xn] is a regular chainif it is triangular and for all i the initial of si:

lc(si,mvar(si))

is invertible with respect to the set:

S|mvar


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Chapter 2


October 10th 2011

Seminar conducted over Skype with Marc Moreno Maza at UWO

2.1 Skype Conversation

Introduced to MMM. All of our paperwork has been submitted referring to DeveloperMaple, now just waiting for Jurgens approval.

Discussion of specifications for computer to run R-Maple:

All of UWO use unix/linux, various flavours of Ubuntu (11.04) Baths preferred type is Opensuse As long as a linux distribution there should be no problem As developer code, the installation of R-Maple may be a little awkward (manually

add packages and so forth)

Do we need a GPU card? JHD wasnt planning on it but MMM thinks it wouldbe useful. They have GPU code for bivariate polynomial factorization, which has

applications for RegularChains

Which GPU? They use the Tesla 2050 but RB will look at alternatives Regular Maple will work without GPU but if one present it will use it MMM will email masjhd-seminar mailing list about GPU papers

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2.2 Other Important PeoplePeople who have written papers with MMM and JHD:

Rong Xiao (UWO) Changbo Chen (UWO) Yu-zhen Xie (UWO) Bican Xie (Peking University)Also, Jurgen Gerhardt - the man in charge of external collaborations at Maplesoft.

2.3 What is R-Maple?

It is the development version of Maple and at the moment only MMM has it. It will beinstalled on machines for JHD, RB, DJW (and possibly Acyr).

Next week - IGG will discuss Flatness.

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Chapter 3

Seminar 3: G. K. Sankaran

October 17th 2011

Will talk about Flatness - well known to commutative algebraists and algebraic geometers.

Example 3.1. As discussed in Seminar 1 this is what we dont want: a variety consistingof

{(1, 1), (1, 1)} ,as it is not equiprojectible onto the x-axis. In algebraic geometry terms: the fibres abovethe two points 1 and 1 are not the same.

What happens if fibres are of positive dimension? Then counting points wont work!

3.1 Flatness

The general geometric idea is you have a map

f : X Bof algebraic varieties or schemes (if multiplicity matters). What does it mean to saythe fibres of f vary continuously - this is what we are aiming for, but from an algebraicviewpoint. In general we assume B is irreducible (and so is not zero-dimensional) otherwisequestion becomes meaningless.

The topologists approach would be to look at fibre bundles, i.e. make the bundleslocally trivial. That is too strong for geometry!

Example 3.2. X a quadratic form:

X =

x y z M

xy

z

P2

with variable M and the mapf : X det(M).

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Example 3.3 (Simpler Example). LetX :=

x2 + y2 + t = 0

A3And let f map to t. Then we would probably still consider this a good family over thecomplexes - its a conic except when t = 0 where it degenerates to a pair of lines. Eventhough t = 0 has a different fibre it isnt too different.

Example 3.4. A standard construction in geometry:

X :=

y2z = 4x3 + g2xz2 + g3z

3 P2

and we map it to

j = g32, g32 27g23 P1where you should note the second component is the discriminant. Normally this is astandard elliptic curve but again special when j = 0 (nodal) - and again this is generallyaccepted as okay. How much does this bother you?

Unexample 3.5. What aboutX := {xy = 0} A2

which is a point and a line - there is a change in dimension at y = 0. This is notequidimensionals and we certainly wouldnt want to classify it as good.

In practice, equidimensionality is not too far from flatness.

Unexample 3.6. A blow-up is constructed by asking at one point in a plane where are

you looking:A2 P1 X := {xi yj = xj yi}

and we map to x A2.Definition 3.1. Let A be a commutative ring and M an A-module. We say that M is

flat over A if MA is exact.That means that if

N1 N2(that is, an injective map) then

M

A N1

M

A N2.

This is a rather cumbersome definition, depending on all A-modules. In fact, it isenough to check:

If a A (and a is finitely generated but usually A is noetherian so that is fine)then

a A M Mis injective (i.e. taking N1 = a and N2 = A).

This is a nice computable thing to check.

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3.1.1 What about maps?We use the definition from Hartsthornes Algebraic Geometry:

Definition 3.2. Let f : X Y be a morphism of schemes. The f is flat over Y at x Xif the local ring of X at x, OX,x, is flat as an OY,y -module with y = f(x).

We say f is a flat morphism if it is flat at every x X.A useful consequence of flatness is the following:

Theorem 3.1. If Y is irreducible and f : X Y is flat then the following are equivalent: every irreducible component of X has dimension dim(Y) + n (this is trivial if X is

irreducible);

at every point y Y, every irreducible component of f1(y) has dimension n.This connects a local and a global property.Another useful theorem demonstrating the power of flatness:

Theorem 3.2. Let T be an integral noetherian scheme. Let X PnT be a closed sub-scheme. For each t T, consider the Hilbert polynomial Pt(z) Q(z) of Xt Pnk(t).

Then X is flat over T if and only if Pt is independent of t.

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Chapter 4


October 24th 2011

Let us head back to the three point example and look at it from an algebraic viewpoint:

How do we define it?

4.1 Grobner Bases

We can define it as an ideal:

F = x2 1, (x + 1)(y2 1) + (x 1)(y 1)

and we can put it into a Grobner Basis algorithm (in this case it doesnt matter whichordering we take) and get

G = {x2 1, (x 1)(y 1), y2 1}.

F is a triangular set and what we would like to do is solve for x, then back-substitute(much like a system of linear equations). However, the degree in y of the second equationdepends on which x we choose. This is dealt with in the Gianni-Kalkbrener Theorem(discovered independently in [Gia] and [Kal87]), which applies when G is a plex GrobnerBasis of a zero-dimensional ideal.

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Theorem 4.1 (Gianni-Kalkbrener Theorem - Grobner Basis version). Let ourvariables be ordered x1 > x2 > > xn and let our plex Grobner basis of a zero-dimensional ideal be G. Write G in the following form:

G =

pn(xn)pn1,1(xn1, xn), . . . , pn1,kn1(xn1, xn)pn2,1(xn2, xn1, xn), . . . , pn2,kn1(xn2, xn1, xn)...

}Bn}Bn1}Bn2...

so that eachBi = G K[xi, . . . , xn] and we sort the elements of |mathbbBi by increasingdegree in xi.

Then given any solution = (k+1, . . . , n), a solution ofBk+1 we define

: K[x1, . . . , xn] K[x1, . . . , xn]xi i

Then:

For all p Bk we have (lcxk(p)) = 0 (p) = 0

If p Bk is the first polynomial (in our ordering) that does not vanish under ,then

(p) | (q) q BkWe also recall one of the nicest properties of Gr obner Bases.

Theorem 4.2 (The Hilbert Basis Theorem). For a Grobner basis G we have:

lm(G) = lm(G)In short, Theorems 4.1 and 4.2 tells you that there are no surprises if you use Grobner

Bases you cannot be lead down a dead end.

4.2 Triangular Sets and Regular ChainsDefinition 4.1. A set of polynomials is a triangular set if it has disjoint main variables(we will order the variables x1 > x2 > > xn). A variable which is the main variableof an element of T we say is algebraic with respect to T. We denote the set of algebraicvariables of T as algVar(T). Any variable which is not algebraic we call a parameter.

Definition 4.2. For a polynomial p with main variable x we define the initial ofp, denotedinit(p) to be

p = init(p) xk + . . .

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Definition 4.3. A triangular set T = {t1, . . . tk} (where mvar(t1) is greater than mvar(t2)and so on) is a regular chain if and only if T := {t2, . . . , tk} is a regular chain; init(t1) is regular with respect to T (i.e. not a zero divisor in T)We have a version of the Gianni-Kalkbrener Theorem in terms of Regular Chains:

Theorem 4.3 (Gianni-Kalkbrener Theorem - Regular Chains version). A zero-dimensional variety can be written as the disjoint union of solutions of regular chains.

This is an easy consquence of Theorem 4.1 by taking the polynomials defined in the

second part of the theorem.Example 4.1. For our three point example we obtain the decomposition:

V(F) = V(x 1, y2 1) V(x + 1, y 1)

4.3 Multiplicity and Positive Dimensions

What about multiplicity? Theorem 4.1 preserves multiplicity; this is due to the fact aGrobner basis generates the ideal rather than the radical of the ideal.

What can we say about multiplication in Theorem 4.3?Can we extend to positive dimensions? The key paper is [GT01] which will be covered

by JHD next week.

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Chapter 5


October 31st 2011

Some useful visualisations provided by Russell Bradford: Figure 5.1 gives a Maple plot ofour interesting example:

{x2 1, (x + 1)(y2 1) + (x 1)(y 1)} (5.1)and shows quite nicely the three points in question (plus a point at infinity).

Figure 5.1: Plot of the curves

If we then run a Grobner basis algorithm we get {(x2 1), (y2 1), (x 1)(y 1)}which, when plotted with Maple in Figure 5.2 is nicely symmetric.

Marc Moreno Maza unfortunately couldnt be here this week but did comment thathe likes the paper to be discussed.

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Figure 5.2: Plot of the Grobner Basis

5.1 Degree Reduction Under Specialization

We will be discussing the paper [GT01].

Notation. Let R be an integral doman (think of k[y1, . . . , yn]), : R

R a homomor-

phism (that we extend by (x) = x to R[x] R[x]).If f R[x] and (f) = 0, define

(f) = deg(f) deg((f))

Notation. Let s = deg((g)). Write

g = G0 + xsG1

where deg(G0) < s, then (G1) R \ {0}.Claim 1. If g

I with (g) < (f) then there exists f

I with (f) < (f) and

(f) = c (f) for some c R \ {0}.(Aside: May be worth looking at altering to not necessarily land in R, i.e. : R S.

However we may need (R) an integral domain.)Let s := deg((g)), so (g) = deg(g) s. Let m := deg(f) (g), then m > 0 since

(g) < (f) deg(f). Write

f = F0 + xmF1 deg(F0) < m

Then from the definitions we immediately have:

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1. deg(G1) = deg(g) s = deg(f) m = deg(F1);2. (F1) = 0 as (g) < (f)

3. (G1) R \ {0}Lemma 5.1. For f, g R[x] with (g) < (f), if deg(g) deg(f) then there existsf f, g such that

1. deg(f) < deg(f);

2. (f) = c(f) for some c R \ {0} (we will see c = (G1));

3. (

f) < (f).Proof. We have a key construction (compare to S-polynomials in Buchberger): Let r =deg(f) deg(g) 0. Then

f = f G1 gF1xr (5.2)= (F0 + x

mF1)G1 xr(G0 + xsG1)F1 (5.3)

The F1G1 terms cancel and the rest of the Lemma follows with some algebraic manipula-tion.

What if deg(g) > deg(f)? We have to alter g rather than f.

Lemma 5.2. If deg(g) > deg(f), then there exists g f, g such that1. deg(g) < deg(g);

2. (g) < (f).

Proof. Let p = deg(g) deg(f) > 0, and define

g = f G1xp gF1. (5.4)

Once again the F1G1 terms cancel and the result follows with some algebraic manip-

ulation.

Theorem 5.1. Let f, g R[x] and (g) < (f). Then there exists f f, g such that1. (f) (g)2. (f) = c(f) for suitable c (R) \ {0}.

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Proof. First construct

f f, g with (

f) = c(f) and (

f) < (f).If deg(g) deg(f) use Lemma 5.1. If deg(g) > deg(f) use Lemma 5.2 repeatedly untildeg(g) deg(f), then use Lemma 5.1.

Now if (f) (g) were done. Otherwise, repeat with the pair {f , g}.

For an ideal I, define

I = min{(h) | h I, (h) = 0}.

Corollary 5.2. For IR[x], : R R we have

1. For any p (I) there exists q I and c (R) \ {0} such that(q) = cp

and(q) = I

2. Let K be a field contained in R such that (R) K and |K= id. Then we cantake c = 1 in the above result. Therefore ker() is a maximal ideal.

Proof. Let g I such that (g) = I, f I such that (f) = p. If (f) = I done, elseuse Theorem 5.1 on {f, g}. This proves 1.

In 2, c K\ {0}, so take g to be c1

q.(Aside: We assume : R R. But if : R S we simply replace c1 by anyelement of R that maps to c1 and the proof also follows.)

In particular, in part 2 of Corollary 5.2 if there exists g I with (g) = 0, then anyp (I) is the specialisation of f I with deg(f) = deg((f)).

The rest of the paper will be discussed next week: the aim of the paper is to proveTheorem 2.1 - for a Grobner basis G of I then x(G) is a Grobner basis of x(I).

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Chapter 6


November 7th 2011

Acyr Locatelli had a question: In Gianni-Kalkbrener when we look at the regular chainswe say the Grobner Basis gives us the decomposition into regular chains. Does it matterwhat order we use for the polynomials? It seems we have to reverse the order in thesecond theorem to get the answer we want. If not, for the three points example we get aprojection on to a different axis.

From week 4 we get projection to the y-axis not the x-axis. Russell Bradford mentionedthat you sometimes have to be fluid with the orderings.

If we take

G := {(x2 1), (x 1)(y 1), (y2 1)} plex(x > y)

we get two solutions for y:y = 1, x2 1y = 1, x 1

RJB commented that Maples RegularChains package gives the three points as distinctregular chains. This is surprising as you can decompose into regular chains just as

Ty := {x2 1, y 1} {x 1, y + 1}

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(for projecting onto the y-axis).The other regular chain decomposition for projecting onto x would be

Tx := {y2 1, x 1} {y 1, x + 1}

by symmetry.So G is set-wise the same if we use plex(x > y ) or plex(y > x) but we get different Pns

in Gianni-Kalkbrener - their triangular decompositions can look very different. In effectwe are choosing which end to chip off from.

6.1 Degree Reduction under Specialization

We will continue discussing the paper [GT01]. The key result from section 1 (last time)was:

Corollary 6.1. For IR[x], : R R we have1. For any p (I) there exists q I and c (R) \ {0} such that

(q) = cp

and(q) = I

2. Let K be a field contained in R such that (R) K and |K= id. Then we cantake c = 1 in the above result. Therefore ker() is a maximal ideal.

What can we learn from this?

6.2 Grobner Bases under Specializations

Consider K[x, T] where T = (t1, . . . , tr) with a fixed block ordering >; we are prioritizingx then we have some ordering on T.

Theorem 6.2. Let IK[x, T] and G = (g1, . . . gs) a Grobner basis for I with respect to

>. Let Kr and assume(I) = 0.

Let gm be the smallest (wrt >) element of G such that (gm) = 0.Then (gm) generates (I) and I = (gm).In particular, (G) is a Grobner basis of (I).

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Proof. Let p be a generator of (I) (as we are essentially in K[x] a P.I.D). By Corollary6.1 there exists q I such that (q) = p and (q) = I.G is a Grobner Basis for I and let H = {gi | degx(gi) degx(q)}. Then q H

(as G is a Grobner Basis ). Since (q) = 0 there must exists at least one gi H with(gi) = 0. In particular gm H (else all these gi would have to vanish).

We also knowdeg((gm)) deg((q)) = deg(p)

as (q) = p generates I.So

I (gm) = degx(gm) deg((gm)) degx(q) deg((q)) = q = Iand hence all inequalities are in fact equalities.

And so deg((gm)) = deg((q)) and hence (gm) generates (I).

This is essentially the Gianni-Kalkbrener Theorem . At each stage of the Gianni-Kalkbrener Theorem we have one privileged variable - this is the inductive step in Gianni-Kalkbrener .

Proposition 6.3. Let Ltx(I) denote the ideal (in K[T]) generated by the leading termsof I with respect to x. Then

I = 0 Lt((I)) = (Ltx(I)).Proof. If I = 0 then degx(gm) = deg((gm)). Then

Ltx((I)) = lt((gm)) = (ltx(gm)) (Ltx(I)) Ltx((I)).

6.3 Applications

Proposition 6.4. Let f1, . . . , f k be a set of polynomials in K[x, T]. Let G = {g1, . . . , gs}be a Grobner Basis of I =

f1, . . . , f k

with respect to a block order >. Let

Kr and

assume (I) = 0. If gm is the smallest (with respect to >) polynomial in G such that(gm) = 0 then

gcd(f1(x, ), . . . , f k(x, )) = gm(x, ).

So once you have your Grobner Basis , however you specialise your GCD is alwaysgoing to be one of the gis; in fact the smallest one that doesnt vanish.

Can restate in terms of comprehensive Grobner Basis :

Corollary 6.5. A Grobner Basis of an ideal I in K[x, T] with respect to a block order forx, T is a comprehensive Grobner Basis in K[T][x], i.e. considering the Ts as parameters.

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We also haveTheorem 6.6. Let I be an ideal in K[x, T] and let G = {g1, . . . , gs} be a Grobner Basiswith respect to a block order >. Let I1 = I K[T] be the first elimination ideal. Let V(I1) and assume (I) = 0. Denote by gm the smallest (with respect to >) polynomialin G such that (gm) = 0. Then if a K we have

(a, ) V(I) (gm)(a) = 0.

We can essentially regard this as the Gianni-Kalkbrener Theorem again. If you havea zero of gm then can extend to (a, ) V(I). So all you need look at are the zeroes ofgm - you can extend one variable at a time. Even when I is not zero-dimensional we can

do Theorem 6.6. We have also not assumed that K is algebraically closed.Example 6.1. Consider K[x,y,a,b] with ordering x > y > a > b. Let

f1 := ax2 + x + y; f2 := bx + y

then we get a Grobner Basis for I = f1, f2 consisting of the blocks:

B1 := {ax2 + x + y,axy by + y,bx + y}, B2 := {ay2 + b2y by}.

Note that these have no polynomials in just a or b. Ifdwe substitute in a = b = 0 (i.e. set = (0, 0)) then the block B2 completely vanishes. Nevertheless (B1) is now

{x + y ,y ,y}.

So therefore x = 0, y = 0.This is a counterexample to the obvious extension to Gianni-Kalkbrener (to non-zero-

dimensional varieties) because we would normally look at block B2 but it actually doesnttell us anything about y. So in fact there is more y-action going on then revealed byB2 in this case. Over a = b = 0, B1 now has 2 variables in it. In some sense at thispoint the Grobner Basis approach has to go back and recompute a basis for B1 at thatspecialization. It isnt that difficult but it tells us things about both y and x. WhereasGianni-Kalkbrener would say each block would tell us all about a single variable. So wecant extend Gianni-Kalkbrener to positive dimensional varieties.

What should we do instead?Over = (0, 0), (B2) = 0 and (B1) = {x + y, y}. Now let B1 = x + y and B2 = y

and we can now apply Gianni-Kalkbrener as we are in a zero-dimensional ideal (althoughin general we may have to recompute a Grobner Basis and so on).

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Chapter 7

Seminar 7: N. Vorobjov

November 21st 2011

7.1 Semi-monotone sets, monotone functions and maps

The project is about Cylindrical Algebraic Decomposition and computing them using someversion of elimination theory (i.e. triangular sets) and today we will be approaching it inan orthogonal way avoiding all algebra.

To start, we will look at cylindrical cells.

Definition 7.1. A 0-dimensional cylindrical cell is just a single point. By induction ifyou have a cylindrical cell C in Rn then a cylindrical cell in Rn+1 is either

graph of a continuous function on C; or the open region between graphs.There is a theory that if you have any definable set (or more specifically a semi-algebraic

set) it can be partitioned into a finite number of cylindrical cells. This decomposition is

Figure 7.1: The cell is either a graph or the open region between the graphs

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dependent on the order of coordinates. Cylindrical cells are supposed to be nice geometri-cally, but really the only obvious nice property is that it is a topological cell: homeomorphicto an open ball of appropriate dimension. Moreover, if it is definable set it is definablyhomeomorphic. Otherwise, cylindrical cells are not particularly that good.

For example the intersection with {xi = c} or {xi < c} may not remain a cylindricalcell, or even connected. Also the projection of a cylindrical cell is not necessarily a cylin-drical cell. Also a permutation of coordinates may not necessarily give a cylindrical cell.Possibly the most important failing is that the cylindrical cell is not necessarily a regulartopological cell.

Definition 7.2. A definable set X is (topologically) regular if (X, X) is homeomorphicto ([0, 1]n, (0, 1)n]

Example 7.1. As shown in Figure 7.2, the boundary of a circle with a point removed is acell but is not regular. The same is true of an open circle with a cut.

We want to fix this. We want a generalisation of a cylindrical cell but where all of theproperties just mentioned would work nicely.

For example, in Figure 7.3 we break the cell so that the monotonicity remains constantin each region. What happens if we generalise this and look at retaining monotonicity?

An O-minimal structure is almost the definition of a structure that admits a CAD sothese may prove to be useful.

In [Dri98], Van den Dries introduces regular structures and proves the existence ofregular cylindrical cell decomposition - by which he means along every coordinate axis if

two points belongs to the set then the interval between them is also contained, as shownin Figure 7.4. He also defines regular functions, a generalisation of monotonicity requiringmonotonicity in each variable when the others are constant.

It is claimed in Van den Dries that definable sets can be decomposed into these regularcells. Unfortunately the word regular is rather misleading as a regular Van den Dries cellis not a topological cell and it doesnt even have the nice property we want.

Example 7.2. The function x21 + x22 on

{x1 > 0, x2 > 0, x1 + x2 < 1} R2

This is shown in Figure 7.5 and is obviously Van den Dries regular (and convex). Thelevel sets become disconnected after a certain point (namely 1/

2).

Figure 7.2: Two examples of cells that are not regular

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Figure 7.3: Splitting a cell to preserve monotonicity of its boundary graphs

Figure 7.4: A regular cell must contain intervals parallel to the coordinate axes

Figure 7.5: An example of a Van den Dries regular cell that does not behave nicely onits level sets

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Example 7.3. An important example showing Van den Dries regular cells are not topolog-ically regular xz

2+y

z

2on the simplex

{x > 0, y > 0, 0 < z < 1, x + y < z} .The graph of the function is topologically regular but if you take the closure of the

graph then the closure will not be the closed ball. This is mainly because over the pointz you have a blow-up - the graph becomes vertical.

Now we look at the sets we wish to replace cylindrical cells with.

Definition 7.3. A coordinate cone in Rn is an intersection of some coordinate hyperplanesand some coordinate halfspaces. So you set some coordinates to 0, some greater than 0,some less than 0 and intersect all of these. We can translate coordinate cones to get finecoordinate cones. Examples are shown in Figure 7.6.

The set X is called semi-monotone if the intersection of X with any affine coordinatecone is connected.

Example 7.4. The chevalier cross, star and ladder shown in Figure 7.7 are all semimono-tone. The crescent moon in Figure 7.8 is not as its intersection with the line shown isdisconnected.

In 2-dimensions semimonotone simply means convex with respect to the coordinate

axes.It is necessary and sufficient to just talk about fine coordinate planes (dont need strict

inequalities). The set is semimonotone if intersection with just each {xj = c} produces asemi-monotone set in that plane.

These have some nice properties: the projection of a semi-monotone onto a plane issemi-monotone, permutation of coordinates preserves semi-monotonicity.

What links semi-monotone with cylindrical cells?

Figure 7.6: An example of a coordinate cone (l) and fine coordinate cones (r)

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Figure 7.7: Examples of semi-monotone sets

Definition 7.4. A function f : X R on a semi-monotone set X is a submonotonefunction if {f < c} is semimonotone and f is upper semi-continuous.

A function f : X R on a semi-monotone set X is a supermonotone function if fis submonotone. That is f is lower semi-continuous and {f > c} is semimonotone.Theorem 7.1. The bounded set X Rn is semimonotone if:

if n = 1 then X is an open interval; otherwise

X =

(x, y) | x X, f(x) < y < g(x)where X is semi-monotone, f is submonotone, and g is supermonotone.

This is sort of a cylindrical cell except our top and bottom function are not continuous.

We can require them to be continuous.

Definition 7.5. A function f is monotone if:

f is both submonotone and supermonotone it is monotone in each variable (in the Van den Dries sense)The analogy of semimonotone sets using monotone functions have the nice property

of being regular topological sets.We can generalise even further: suppose we have these semi-monotone sets as analogies

to cylindrical sets. These are open sets, what about their smaller dimension components?

Figure 7.8: This is not a semi-monotone set as the shown intersection is disconnected

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Figure 7.9: Splitting the chevalier cross into the region between a submonotone and su-permonotone graph

Figure 7.10: The same region of graph can be used to define the range and the domainthrough projection

For co-dimension 1 we suggest graphs of semimonotone sets, for positive co-dimensiongreater than 1 we suggest graphs of semimonotone maps - maps defined on semi-monotonesets

X (f1, . . . , f k)where the fi are monotone with certain extra conditions on the fi. The fi have to beconnected between themselves in a very non-trivial way. In the definition of monotonefunctions they have to be Van den Dries in each variable. The analogy becomes combina-torially very difficult but also very important.

The preservation of type of monotonicity is formulated using matroid theory: thereneeds to be a matroid associated with the monotone map and this matroid must stay thesame when you permute the parameters.

In particular the graphs of monotone maps and structures are also regular in a topo-logical sense and they have a nice exchange property - you can change between domainand range and the same graph defines both, which you can see trivially in R2 in Figure7.10.

In dimension 3 (and 2) you can have Cylindrical Algebraic Decomposition with semi-monotone sets as your cells and with continuous top and bottom functions. For dimension

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4 it is a almost true except you have to use certain linear change of coordinates alongthe way.The topics discussed in this seminar were based on [BGV10].

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Chapter 8


November 28th 2011

8.1 Admin

The computer has been ordered for DJW but we still need to hear back about what GPUcard to order. All the legal paperwork has been signed and so we now need the actualsoftware - preferably as close to MMMs set-up as possible.

Applications have closed for the PostDoc. The interviews by JHD and RJB for theshortlisted candidates will mainly be held on 9/12/11: there will be presentations in East

Building in the morning. DJW is to attend and AL is welcome to attend if he is free.

8.2 Computing CAD via TD [CMXY09]

JHD will discuss [CMXY09] which was presented at ISSAC 2009 using the slides they gaveat the conference.

8.3 Background

CAD is a fundamental tool and from the original introduction of the algorithm in 1973 it

has been improved in various ways. These were all centered around the idea of projectingfrom Rn to Rn1, understanding the problem there and using that to understand theproblem in Rn.

The general idea: when you project down you forget an awful lot the booleancombination of the fi and gj gets forgotten. You then decompose R

n1 into cells and thencreate cylinders over each cell. This is shown in Figure 8.1.

Definition 8.1 (In the style of Collins). An order-invariant CAD means over eachcell Cj Rn1 = R[x2, . . . , xn] we have branches of functions Fi(x!, x2, . . . , xn) = 0 whichare delineable over Cj :

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Rn :

BooleanCombination f1 = 0, gj > 0

// Rn : cylinders over each cell

Rn1 : hi // Rn1 : decomposed into cells

OO

Figure 8.1: Idea behind Collins Method

1. each branch has a constant multiplicity over Cj and

2. over Cj there is a fixed ordering of branches F(1)j , F

(1)j , . . .: the x1 coordinate cor-

responding to F(1)j (x1, . . . , xn) = 0 is less than the x1 coordinate corresponding to

F(1)j (x1, . . . , xn) = 0 and so forth.

So the motivation is to understand the link between CAD and Triangular Decomposi-tions.

Definition 8.2 ([CMXY09] take on CAD). You can define a sign-invariant CADinductively where you create sections and stacks over a lower dimension CAD.

Important example to always consider - Tacnode curve:

y4

2y3 + y2

3x2y + 2x4 = 0 (8.1)

8.4 Another view of CADs

You can view a CAD as a partition ofRn such that all the cells are cylindrically arranged(projections of pairs of cells on the first k coordinates are equal or disjoint) and each cellis a connected semi-algebraic set (called a region).

It is taken as obvious (although it is not necessarily all that obvious) that this is equiva-lent to the other definitions of CADs. Note that this definition covers decompositions thatmay not be constructible though Collins methodology see Figure 8.2 . This may proveto be very useful. Also, note that Toms Lemma may be needed to see the semi-algebraiccondition.

After much discussion it seems like it may be necessary to look at the fine-print ofCollins definition to check they match exactly.

For example at a vertical tangent, Collins methodology (and possibly Collins defi-nition) would insist on splitting whereas this definition would not require that to retaincylindricity.

But what about the actual technical definitions [CMXY09] gives? It may be that thisview is just a useful way to think of CADs rather than a technical definition.

For Fn Q[y1, . . . , yn] a CAD ofRn is Fn-invariant if above each region of it, the signof each f Fn is constant.

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Figure 8.2: The decomposition on the left is a CAD in all senses of the definition. Thedecomposition on the right is a CAD according to [CMXY09] but can not be created byCollins method

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Chapter 9


December 5th 2011

9.1 Last Time

We noted that their definition of a CAD was a sign-invariant CAD rather than order-invariant. Is their viewpoint consistent with Collins definition?

Grant Passmore (from Cambridge) may be listening online.

9.2 Continuation of discussion on [CMXY09]Collins original paper [CMA82] defines invariant functions with regard to just their signs.In his delineability definition he talks about multiplicity but does not in his CAD definition.Collins defines CAD as sign-invariant but the definition of delineability is order-invariant.There is different approaches in the CAD community over whether to approach CADsfrom sign-invariance or order-invariance.

To disregard GKS non-example (Figure 9.1) in the informal viewpoint (cylindrically

Figure 9.1: This is a CAD according to the informal viewpoint of [CMXY09] but can notbe created by Collins method

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arranged subsets) you simply need to add the fact that the boundaries are graphs offunctions.

9.3 Their method for making a CAD

InitialPartion MakeCylindrical MakeSemiAlgebraic

This has no features in common with Collins. Begins over C and produces a = 0, = 0decomposition ofCn. They then make it cylindrical (in the C-cylindrical sense). Theythen convert it to a true CAD.

Their constructible sets are based on regular systems, [T, h]. T is a regular chain, so

T is triangular and the initials are invertible modulo the other polynomials. h has to beregular with respect to sat(T). Essentially we set all polynomials in T to be 0 and h tobe non-zero.

Theorem 9.1. Every constructible set can be written as a finite union of the zero sets ofregular systems.

Example 9.1. The constructible set:

x(1 + y) s = 0 (9.1)y(1 + x) s = 0 (9.2)

x + y

1= 0 (9.3)

With x > y > s gives:

r1 :=

T1 :=

(y + 1)x s

y2 + y sh1 := y 2s + 1

r2 :=

T2 :=

x + 1y + 1

sh2 := 1

(9.4)

This is exhibiting a mixed-dimension behaviour but the single point, in some sense, lieson the line.

9.3.1 InitialPartition

This generates an intersection-free basis of the sets f1 = 0, f2 = 0, . . . , fs = 0 andf1 fs = 0.

9.3.2 MakeCylindrical

They then do a comprehensive triangular decomposition want to understand zerosof this regular system allowing anything to happen to the parameters. The procedureSeparateZeros projects the zeros onto the parameter space and then construct zero sets

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where the initials of the polynomials do not vanish and the polynomials are squarefreeand coprime at every individual point in R. This is different to them being squarefree andcoprime over R!

So if you get a point where they are not squarefree and coprime you have to start anew region where, instead of f and g you have {f/h,g/h,h} where h is gcd(f, g).

We now create a cylindrical decomposition:

For n = 1 we decompose into regions where each polynomial is zero and one morewhere they are all non-zero.

For n > 1 we lift over a cylindrical decomposition D = {D1, . . . , Ds} by eitherincluding Di

C or a similar construction as in the n = 1 case.

Example 9.2. In the parabola case: r4 is the quadratic non-zero, r3 is the quadratic beingzero and a non-zero, r2 is the quadratic and a being zero and b non-zero and finally r1 iseverything zero.

We get the nice decomposition given in the tree diagram.

9.3.3 MakeSemiAlgebraic

They now use Collins definition of delineable. They use a corollary that if the initial ofeach polynomial in {p1, . . . , pr} does not vanish for any R and the polynomials aresquarefree and coprime at each point then the set {p1, . . . , pr} is delineable over R.

9.4 Example - the parabola

For the parabola this method produces the minimal 27 cells (why minimal? Look at thetree diagram of the complex case, noting every = case produces 2 cells when working overthe reals).

The original projection operator in [CMA82] produces p, b2 4ac, c, b, a (none ofwhich are surprising weve seen them all in this method) but yet the reconstructionproduces 115 cells (around 4 times the number this method produces).

It might be that they consider some of the other polynomials in cases where we ignorethem? It is worth noticing the Brown-McCallum operator also produces the minimal 27

cells, but may fail in rare cases.In this example it produces the same number of cells as the Brown-McCallum operator,surely the same cells. Is there any way to say that the Triangular Decomposition methodis at least as good or better than Brown-McCallum.

If we dont insist on being cylindrical, what do we get? How much does it cost us?

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Chapter 10

Seminar 10: A. Locatelli

December 12th 2011

10.1 Original Definitions

Assume throughout that R is a real closed field. We define a Cylindrical Algebraic De-composition via induction:

Definition 10.1. For R1: a CAD of R is a a sequence (S1, . . . , S k). If k = 1 thenthe decomposition is just the whole of R. If k > 1 then exists real algebraic numbers

1, . . . , k. Let S1 = (, 1) and letS2i = {i} (10.1)

S2i+1 = (i, i+1) (10.2)

and finally let S2k+1 = (k, ).Now for Rn assume we have (S1, . . . , S k) a CAD ofR

n1. For each i we have continuousreal-valued functions on Si to R:

fi,1 < < fi,i (10.3)

If i = 0 then Si,1 := Si R (the whole cylinder). Otherwise

Si,2j = {(a, b) | a Si, b = fj(a)} (10.4)

Si,2j+1 = {(a, b) | a Si, fj(a) < b < fj+1(a)} (10.5)We call the Si,j s the cells of the CAD.

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3 4

Ralg

Figure 10.1: Partition ofRalg

10.2 Alternative View

We have the alternative view.

Proposition 10.1. Given a CAD (as defined above) then it partitions Rn and the cells

are cylindrically arranged.This is easy to see (it follows straight from the definition). We also wish to say each

cell is a connected semi-algebraic set. Is this true? Look at this example:

Example 10.1. We use 3 and 4 to split Ralg as shown in Figure 10.1. Then (3, 4) can besplit as (3, )(, 4) which are both open in Ralg so (3, 4) is not topologically connected.This suggests we need semi-algebraically connected, as just connected doesnt reallymake sense here.

Over R semi-algebraically connected is exactly the same as Euclidean connected.Semi-algebraically connected: you cant write it as a disjoint union of semi-algebraic

open sets.

Does this affect our alternative viewpoint? Probably not as they probably only meanit for the reals.

10.3 Chen et als paper [CMXY09]

Tries to generalise CAD to other fields. They introduce constructible sets which havesimilar properties to semi-algebraic sets.

Definition 10.2. A locally closed set is a set that is the intersection of an open and closedset.

A constructible set, C, is a finite union of sets L1, . . . , Ln where each Li is locally

closed.If we are in a Zariski topology then any constructible set can be written as a finite

union of sets of the form (Ai \ Bi) where the Ai and Bi are algebraic varieties. This ispretty much the closest we can get to semi-algebraic - we allow equality and inequalitybut have no ordering.

Let C be a constructible set and let P k[u, yn]. For f P view f as in k[u][yn] andassume deg(f) = m.

Definition 10.3. P separates over C if for all C:

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1. deg(p(, yn)) = m2. p(, yn) are square-free and coprime

Definition 10.4. Let Kn be algebraically closed. We want to define a cylindrical decom-position by induction.

K1: a CD is a finite collection of sets {D1, . . . , Dr+1} where if r = 1 define D1 = Kand if r > 0 we assume we have p1, . . . , pr k[y1] and define

Di = V(pi), Dr+1 =

y1 |

pi(y1) = 0

(10.6)

Kn: Assume we have

{D1, . . . , Ds

}which is a CD for Kn1. Fix i and let pi,1, . . . , pi,i

separate over Di. If i = 0 define Di,1 = Di K. If i > 0 defineDi,j = {(, yn) | Di, pi,j(, yn) = 0} (10.7)

Di,s+1 =

(, yn) | Dr+1,

j

pi,j(, yn) = 0 (10.8)

10.4 F-invariance

For CAD we have a notion of F-invariance each function in F has constant sign overeach cell.

For CDs we have the notion that a decomposition is an intersection-free basis of theconstructible sets V(fi) and {y Kn | f1(y) fs(y) = 0}.

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Part II

Semester 2 20112012

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Chapter 11


February 10th 2012

Apologies from DJW and AL. ME listening in over the recording.JHD talking about his thoughts on CMMXY slides. Collins original algorithm worked

by projecting to R1, dividing and lifting. The U.W.O. algorithm is based on the rather dif-ferent algorithm by decomposing Cn depending on if the polynomials are zero or vanishingnowhere (analogue to constant-sign in R). They then makes this set C-cylindrical (projec-tions are equal or disjoint). Finally they make it semi-algebraic to produce a sign-invariantCAD ofRn.

First thought last semester NNV introduced the idea of a monotone set. That is, notjust cylindrical in the normal sense but cylindrical with respect to all possible projections.

Looking at Figure 11.1 we see that a cylindrical decomposition projecting onto x wouldresult in an absolute minimum of 25 cells: 9 two-dimensional cells, 12 one-dimensionalcells and 4 zero-dimensional cells (i.e. points). If instead we wish to have a monotonedecomposition we must add in many more cells (shown in Figure 11.2).

1

3

5

3

1

3

5

3

1

Figure 11.1: CAD of two circles

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Figure 11.2: Monotone CAD of two circles

So a question JHD has asked himself is Is it possible to replace MakeCylindricalwith MakeMonotone in the UWO algorithm?. GKS noted there was an obvious way todo this - decompose in one direction, then in another and so on. But you might mess upthe first direction. But you are simply refining the partition in each step so GKS thinksit shouldnt affect anything from earlier steps. So JHD and GKS both feel it should work.JHD noted that the initial step (MakeCylindrical) is dependent on an underlying variableorder to form the triangular partition which may be affected by looking at monotone sets.

RJB questioned how useful monotone sets are. JHD couldnt recall any exact ap-plications. Later NNV pointed out that his group initially looked at monotone sets fortriangulation of one-parameter definable families, which is part of a program to prove their

conjecture with Gabrielov about homotopy equivalence of definable sets and their compactapproximations. This is briefly outlined in their introductory papers on semi-monotoneand monotone functions.

JHD also did not know if there were currently any ways to compute monotone decom-positions. NNV later pointed out that there was an algorithm for R4 but this was createdvery early on so should definitely be able to be improved.

JHDs second thought was as follows. Consider two non-intersecting circles in R2,whose projections on the x-axis also do not interset. These could either be arbitrary oraligned. Then the circles can be inputted into CylindricalAlgebraicDecompose .

If we look atL := [x2 + y2

1, (x

3)2 + (y

1)2

1] (11.1)

(offset circles at (0, 0) and (3, 1)) using a polynomial ring PoynomialRing(y, x) we get 25cells and it corresponds to Figure 11.1.

However, if I look instead at

L := [x2 + y2 1, (x 3)2 + y2 1] (11.2)

(the non-offset version) things are subtly different in the middle. There is a spurious splitat x = 32 . What happens is that the complex conics of the two circles meet above

32 . One

could say this was unavoidable but instead let us look further into it.

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Using CylindricalDecompose we work over C. Taking the unaligned cirles it splits atthe critical points in R but also the point with complex intersection with complex roots. Inthe aligned version the two points where the curves intersect coincide and become partlyreal.

So now when we use MakeSemiAlgebraic it ignores this branch in the unaligned versionas there are no real x roots, whereas in the aligned version there is a solution for x real soit does not ignore it.

So the output is near-identical working over C, but MakeSemiAlgebraic doesnt realisethat the real x = 32 is in some sense spurious. But what does spurious mean? The realgeometry just to the left of 32 , at

32 and just to the right is identical (but the complex

geometry is not). So this is real-spurious but not complex-spurious (in some undefined

sense). So is there a way to remove these points between steps two and three?Does this happen often? Could you do some damage by removing these points? RJBpointed out that in real-word applications you are much more likely to have naturalsymmetries so these probably do occur often.

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Chapter 12

Seminar 12: D. J. Wilson

February 17th 2012

(Notes from J. H. Davenport)See http://www.cs.bath.ac.uk/~djw42/triangular/index.html , which is where

this work lives. There are three files.

Example Text file The first section comes from [CMXY09]. The second is the branchcut examples from [Phi11]. The third are motion planning examples [Dav86], withmore to come. Fourth section is miscellaneous, which will need more categorisa-tion.

We store the Tarski formula (if one), free and quantified variables, suggested order,best achieved number of cells, and any notes, including a reference to first occurrenceDJW can find.

Maple file This is a single file (at the moment). This is, however, split up into procedures,e.g. CMXYExamples and matching CMXYExamplesInfo. For example

A,B:=CMXYExamples(16);

CylindricalAlgebraicDecompose(A,PolynomialRing(B));

(in other words, the variable order returned is the correct one for Maple). Note also

that

L:=CylindricalAlgebraicDecompose(A,PolynomialRing(B),output=list);

nops(L);

returns the number of cells.

QEPCAD The QEPCAD file can be cut/pasted into QEPCAD. JHD suggested that wemight explore with Chris Brown whether there were better ways.

Help on running these commands in Maple is also linked from the URL above.

43
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12.1 Discussion Note that there is an abstract problem, several possible Tarski formulations, e.g.

the nave one, various forms of preconditionings, and various orderings (compatiblewith the quantifiers if any).

The example of last week shows that we should distinguish between the cells achievedby an algorithm (27 in the case of two aligned circles) and the minimum possible forany cylindrical decomposition (25 in this case). However, it is not obvious how tocompute the latter in general.

Some of the examples comes from [BG06], which gives rise to both the originalproblem and the semi-manually-simplified version. Could we aim to mechanise moreof this?

See also [Kau11] for an interesting an easily visualized problem involving a circleand intersecting hyperbola.

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Chapter 13

Seminar 13: Informal Discussion

February 24th 2012

Discussion regarding paper submission to Calculemus at CICM 2012 from JHD, RJB andDJW.

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Chapter 14

Seminar 14: Informal Discussion

March 2nd 2012

Discussion regarding paper submission to Calculemus at CICM 2012 from JHD, RJB andDJW.

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Chapter 15

Seminar 15: James H. Davenport

March 9th 2012

GKS sends apologies, not feeling well.ME starts 16th April. Shall try to get him the same box as DJW.CICM paper has been submitted what are the outstanding research questions?

Quite a few.DJW came up with TNoI as a tiebreaker.As JHD sees it, one way to look at the big roadmap as a hierarchy (see Figure 15.1):

Very abstract problem the logical problem semi-algebraic variety equations for SAV (formulation) oriented formulation (have put an order on the variables) (Partial) Cylindrical Algebraic Decomposition (possibly via Cylindrical Decomposi-

tion over the complexes)

How to move from oriented formulation to CAD? There is Collins, Collins-Hong, UWOand possibly partial UWO.

How to move from formulation to oriented formulation? There is DSS, greedy DSSand Brown heuristic.

CICM paper covers moving semi-algebraic variety to formulation and is alongsideNalinas methods. To measure this we have DSS and TNoI. CICMs punchline is that

TNoI is better than DSS at this stage. In CICM we had already an underlying orderingswhen reformulating. Next up some data collection of different orderings and different for-mulation. A research aim is to merge Nalinas methods and the CICM methods - we havethe idea behind this but need to construct an example to test.

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Figure 15.1: Overall roadmap of the various techniques

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We have a partially analysed case to cover with TNoI replacing a polynomial with afew smaller polynomials but lowering TNoI. Worth working a couple of worked examplesto see if there is a distinct drop in resultants.

JHD paper currently circulating. An attempt to say that when people are doing se-mantics they need to worry about algebra too. One example is injectivity in the Joukowskimap. One problem with injectivity is that the casting of the problem requires double thenumber of variables thinking of it as

z z f(z) = f(z) z = z . (15.1)Brown reformulates it as

z z x = x f(x) = f(x) (15.2)and uses other features of Qepcad to prove this for Joukowski. Other tactics might beto throw analysis at the problem - is this the right way to look at injectivity? Can wethink in some kind of hybrid omplex-real viewpoint. We have the power of CylindricalDecomposition over C.

Another abstract problem we have at the moment is connectivity. We know that CADis in some sense not the right approach as it is doubly exponential. There are knownsingly exponential algorithms. There is a paper by Hoon Hong from the early 90s wherehe says there are singly exponential algorithms but the constants are so big it takes yearsto prove.

Next meeting: DJW away next week. 3 more weeks of term. Provisionally we shall bemoving to 3:15 on Thursdays, starting on the 22nd March.

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Chapter 16

Seminar 16: Christopher W.

BrownMarch 22nd 2012

Conducted with Christopher Brown over Skype.Original Joukowsky transformation email: began with the straightforward transforma-

tion quantified over a,b,c,d. Just trying to calculate directly exhausts the PrimeList.First we negate to get existential. Theres nothing fundamental in the choice of ex-

istential other than the software Brown used to reformulate the logical statement. The

other reason is Qepcad using equational constraints.Collins, McCallum and Brown found the following. If you have equations implied by

your input then you can reduce the projection in CAD building because you really onlyrequire that the sections form those equation polynomials are delineable and the non-equational restraints are (in some sense) sign/order invariant on the equations (what theydo off the equations doesnt matter).

QEPCAD can take advantage of these constraints but they have to be declared explic-itly. Have to use prop-eqn-const mode so QEPCAD uses a special projection that cantake advantage of these constraints. The eqn-const-poly sets equations to do this. ScottMcCallum, ISSAC 2001 (On Propogation of Equational Constraints in CAD-Based Quan-tifier Elimination, [McC01]) showed this works with certain caveats. QEPCAD checks

these conditions hold. No paper talks about the move from that paper to the implemen-tation in QECAD.

Its not immediately obvious how to look at a formula and pick out these constraintsas they might be masked. Hence QEPCAD cant find them automatically, so users have todeclare them explicitly. This reduction in projection can have huge savings in the lifting.

Why remove c and d? This is due to an implementation limitation in QEPCAD there are certain criteria that have to be satisified for the reduced criteria to be used. Ifa polynomial is not one of these equational restraints, its discriminant does not have tobe added to the projection assuming the discriminant would not be zero in any positive

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dimensional cell. So in determining whether a polynomial is not zero on positive cells andso forth you dont want to just consider the polynomial by itself as other constraints mightimply that it is non-zero. The way QEPCAD is written, the olny constraints it uses forthis is those listed as assumptions and these have to be on free variables not quantified.So we leave c and d free to allow c2 + d2 1 > 0 as an assumption.

This will be improved in later versions of QEPCAD you wont need to free variablesor declare assumptions.

How can we measure the size of the projection set? Before lifting you can typed-proj-f to list all of the projection factors. Similarly we can project with the assump-tions and run d-proj-f to compare. Also, d-stat will give a summary including numberof projection factors.

For Joukowsky there are 187 projection factors at level 1 without constraints, but ifyou do declare the constraints there are only 22 factors.As well as the two constraints, it also uses their resultant as a constraint. So you get

to use the resultant as a constraint in a lower-level projection space.In some sense what the aim of equational constraints is not dissimilar to what Lazard

et al wanted to do with the discriminant variety or the UWO algorithm. It is trying to useequations differently in the projection stage. There is also the Brown-McCallum paperfrom 2005 ([BM05]) but this isnt implemented (as there are more caveats). Also a 2009paper ([MB09]).

With the discriminant variety method you project away all bound variables in one step.There is no lifting cell by cell (so no explicit structure like in CAD). They provide the cell

decomposition of free-variable space so that (roughly speaking) there are no surprisesover each cell. You cant quite talk about delineability in the same sense (as there isno real definition - 2009 paper tries to tackle this) but there is a notion that fibres arewell-behaved (none are created or destroyed and none intersect). Where that breaks down- what happens on the discriminant variety itself? And when you have systems that arenot generically zero-dimensional as you would project and the variety would be too bigin some sense.

Brown has some software that can take an input problem and suggests a good way tophrase it for QEPCAD. Suggests possible equational restraints, variable ordering, declaresequational constraints and assumptions using the correct syntax. Brown will try to packagethis up to send out.

Overall there is the issue of lots of underlying decisions before you even input a probleminto an algorithm. For example, should you split? It may not be obvious that splittinga problem is useful but there may be second-order interference that you benefit fromeliminating.

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Chapter 17


March 29th 2012

Discussion regarding rebuttals for referees comments on paper submission to Calculemusat CICM 2012 from JHD, RJB and DJW.

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Chapter 18


April 26th 2012

Apologies from GKS.Introductions for Matthew England: GKS and AL. They are concerned with theoretical

side which is interesting as cylindricity over C, as far as we know, hasnt really been lookedat in a non-practical setting.

Whats currently going on: DJW giving Department PGR talk tomorrow. ThenDJW will talk at SIAM and CAIMS in June including a trip to UWO and Maplesoft. InJuly two major conferences: ISSAC (the major Computer Algebra conference) for which

we have applied for a poster and CICM (Calculemus track) for which we have had a fullpaper accepted for. JHD is also running an OpenMath workshop there and helping withthe Doctoral Programme. DJW is applying to take part in the Doctoral Programme.

Looking at EPSRC Application and travel budget. Weve got a reasonable amount ofmoney for travel in the budget but it depends on who goes where. DJW has got a grantfor one of the Canadian conferences and hopefully for CICM.

Masterplan for travel:

Canada DJW

ISSAC JHD, ME, DJW

CICM JHD, DJW

By next meeting will have hopefully finalised travel plans so we can cover early regis-tration. Probably doesnt warrant a trip for ME to Canada just yet (especially as UWOwill be at ISSAC) so may be better later in the year.

DJW update: VPN isnt working and theyre looking at other solutions. JHD pointsout that he can access UWOs SVN for paper writing. Might be worth asking if it ispossible to route through UWO. RJB points out it is worth checking if is there any wayto get a snapshot up and running so we can look at the source code. JHD will talk withMMM at UWO to see how they deal with it and if there is a temporary workaround.

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Looking at grant proposal there is a key paragraph:

Adding Laziness to cylindricity. Is it possible to adapt the MakeCylin-drical step of the algorithm of [CMXY09] so as only to provide cylindricitycorrespoinding to the blocks in [the quantifier elimination problem]? If so, thiswould be a significant step forward for practical quantifier elimination, andpossibly bring the complexity of the algorithm closer to the theoretical lowerbound of O(N2

a). I would certainly be of great advantage to the other appli-cations of a CAD that dont need cylindricity as such, including applications[robotics motion planning] and [analysis of branch cuts].

([Dav11, Case for Support, pg 6])

Consider two 1-dimensional curves meeting in two 0-dimensional points. The twocurves are f(x, y) = 0 and g(x, y) = 0 with h(x) = 0 defining where the projectionshit the x-axis. Then h(x) = 0 f(x, y) = 0 g(x, y) = 0 describes the two curvesapart from genuine and spurious intersection points. This is the full-dimensional part ofthe triangular representation. There is then one or two extra sets, probably one for thegenuine intersection points and one for the spurious intersection points. Lazy triangulardecomposition ignores these lower dimensional problems and wraps them up as a problemsimilar to the input.

The key question is: Is it possible to add laziness to cylindricity to get awayfrom the doubly-exponential bound? This is where we want to be headed so needto look at how laziness works - ISSAC 2010 [CDM+10] introduces laziness and the ideaof a border polynomial; ISSAC 2011 [CDM+11] then improves the idea of the borderpolynomial.

ME has spent the last few days looking at CAD and has worked on Collins algorithm.3rd May: JHD gives his apologies. DJW, ME may have an informal meeting. No need

for recording. Next meeting: 10th May

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Bibliography

[BG06] Christopher W Brown and Christian Gross. Efficient Preprocessing Methodsfor Quantifier Elimination, volume 4194. Springer Berlin Heidelberg, Berlin,Heidelberg, 2006.

[BGV10] Saugata Basu, Andrei Gabrielov, and Nicolai Vorobjov. Semi-monotone sets.arXiv.org, math.LO, April 2010.

[BM05] Christopher W Brown and Scott McCallum. On using bi-equational constraintsin CAD construction. In ISSAC 05: Proceedings of the 2005 internationalsymposium on Symbolic and algebraic computation. ACM Request Permissions,July 2005.

[CDM+10] Changbo Chen, James H Davenport, John P May, Marc Moreno Maza, BicanXia, and Rong Xiao. Triangular Decomposition of Semi-algebraic Systems.arXiv.org, cs.SC, February 2010.

[CDM+11] Changbo Chen, James H Davenport, Marc Moreno Maza, Bican Xia, andRong Xiao. Computing with semi-algebraic sets represented by triangulardecomposition. In ISSAC 11: Proceedings of the 36th international symposiumon Symbolic and algebraic computation. ACM Request Permissions, June 2011.

[CMA82] George E Collins, Scott McCallum, and Dennis S. Arnon. Cylindrical AlgebraicDecomposition I: The Basic Algorithm. Computer Science Technical Reports,1982.

[CMXY09] Changbo Chen, Marc Moreno Maza, Bican Xia, and Lu Yang. ComputingCylindrical Algebraic Decomposition via Triangular Decomposition. In ISSAC09, pages 95102, Seoul, Republic of Korea, 2009. ORCCA, University ofWestern Ontario.

[Dav86] James H Davenport. A Piano Movers Problem. ACM SIGSAM Bulletin,1986.

[Dav11] James H Davenport. EPSRC Proposal. pages 132, January 2011.

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[Dri98] L. V. Dries. Tame topology and o-minimal structures. London MathematicalSociety lecture note series. Cambridge University Press, November 1998.

[Gia] P. Gianni. Properties of Grobner Bases under specializations. Proc. EURO-CAL 1987, pages 293298.

[GT01] Patrizia Gianni and Barry Trager. ScienceDirect - Journal of Pure and AppliedAlgebra : Degree reduction under specialization. Journal of pure and appliedalgebra, 2001.

[Kal87] M. Kalkbrener. Solving systems of algebraic equations by using Grobner bases.Proc. EUROCAL 1987, pages 282292, July 1987.

[Kau11] M Kauers. How To Use Cylindrical Algebraic Decomposition. SeminaireLotharingien de Combinatoire, 2011.

[MB09] Scott McCallum and Christopher W Brown. On delineability of varieties inCAD-based quantifier elimination with two equational constraints. In ISSAC09: Proceedings of the 2009 international symposium on Symbolic and alge-braic computation. ACM, July 2009.

[McC01] Scott McCallum. Proceedings of the 2001 international symposium on Sym-bolic and algebraic computation - ISSAC 01. In the 2001 international sym-posium, pages 223231, New York, New York, USA, 2001. ACM Press.

[Phi11] Nalina Phisanbut. Practical Simplification of Elementary Functions usingCylindrical Algebraic Decomposition. PhD thesis, University of Bath, 2011.

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Triangular Sets Seminar