The shapes of level curves of real polynomials near strict local … shapes... · WhenevertheoriginisaMorse strictlocalminimumthe small enough levelcurvesareboundariesof convex topologicaldisks

The shapes of level curves of realpolynomials near strict local minima

Miruna-Ştefana Sorea

Max Planck Institute for Mathematics in the Sciences, Leipzig

Algebraic and combinatorial perspectives in the mathematical sciences (ACPMS)

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 1 / 42

Goals

• objects: polynomialfunctions f : R2 → R,f (0, 0) = 0 such that O isa strict local minimum;

• goal: study the realMilnor fibres of thepolynomial (i.e. the levelcurves (f (x , y) = ε), for0 < ε� 1, in a smallenough neighbourhood ofthe origin). f (x , y) = x2 + y 2


Whenever the origin is a Morse strict local minimum thesmall enough level curves are boundaries of convex

topological disks.


Question (Giroux asked Popescu-Pampu, 2004)Are the small enough level curves of f near strict local minimaalways boundaries of convex disks?

Counterexample by M. Coste: f (x , y) = x2 + (y 2 − x)2.


• Problem: understand these phenomena of non-convexity.• Subproblem: construct non-Morse strict local minimawhose nearby small levels are far from being convex.


Question

What combinatorial object can encode the shape bymeasuring the non-convexity of a smooth and compactconnected component of an algebraic curve in R2?


The Poincaré-Reeb graph

associated to a curve and to a direction x

DefinitionTwo points of Dare equivalent ifthey belong to thesame connectedcomponent of afibre of theprojectionΠ : R2 → R,Π(x , y) := x .


The Poincaré-Reeb tree

Theorem ([Sor19b])

The Poincaré-Reeb graph is atransversal tree: it is aplane tree whose open edgesare transverse to thefoliation induced by thefunction x ; its vertices areendowed with a totalpreorder relation induced bythe function x .


The asymptotic Poincaré-Reeb tree

-small enough level curves;-near a strict local minimum.

Theorem ([Sor19b])The asymptotic Poincaré-Reebtree stabilises. It is a rootedtree; the total preorder relationon its vertices is strictlymonotone on each geodesicstarting from the root.

Impossible asymptoticconfiguration:


• Characterise all possibletopological types ofasymptotic Poincaré-Reebtrees.

• Construct a family ofpolynomials realising alarge class of transversaltrees as theirPoincaré-Reeb trees.


Main result

• introduction of new combinatorial objects;• polar curve, discriminant curve;• genericity hypotheses (x > 0);• univariate case: explicit construction of separable snakes;• a result of realisation of a large class of Poincaré-Reebtrees.

Theorem ([Sor18])Given any separable positive generic rooted transversaltree, we construct the equation of a real bivariate polynomialwith isolated minimum at the origin which realises the giventree as a Poincaré-Reeb tree.


Tool 1 : The polar curve

Γ(f , x) :=

{(x , y) ∈ R2

∣∣∣∣ ∂f∂y (x , y) = 0}

It is the set of points where thetangent to a level curve isvertical.


Tool 2 : Choosing a generic projection

Avoid vertical inflections: Avoid vertical bitangents:


The generic asymptotic Poincaré-Reeb tree

Theorem ([Sor19c])In the asymptotic case, if thedirection x is generic, then wehave a total order relationand a complete binary tree.

Two inequivalent trees


Tool 3: The discriminant locus

Φ : R2x ,y → R2x ,z ,Φ(x , y) =(x , f (x , y)

).

The critical locusof Φ is the polarcurve Γ(f , x).

The discriminantlocus of Φ is thecritical image∆ = Φ(Γ).


Genericity hypotheses

The family of polynomials that we construct satisfies thefollowing two genericity hypotheses:

• the curve Γ+ is reduced;

• the map Φ|Γ+ : Γ+ → ∆+ isa homeomorphism.


1. Positive asymptotic snakeTo any positive (i.e. for x > 0) generic asymptoticPoincaré-Reeb tree we can associate a permutation σ, calledthe positive asymptotic snake.


2. Arnold’s snake (one variable)

One can associatea permutation to aMorsepolynomial, byconsidering twototal orderrelations on theset of its criticalpoints: Arnold’ssnake.


2. Arnold’s snake (one variable)

The study ofasymptotic formsof the graphs ofone variatepolynomialsf (x0, y), for x0tending to zero.

Theorem([Sor18])

σ = τ.


ProofThe interplay between the polar curve and the discriminantcurve:

σ = τ =

(1 2 32 3 1

)Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 20 / 42

The construction

SubquestionGiven a generic rooted transversal tree, can we construct theequation of a real bivariate polynomial with isolated minimumat the origin which realises the given tree as a Poincaré-Reebtree?

Theorem ([Sor18])We give a positive constructive answer: we construct afamily of polynomials that realise all separable positivegeneric rooted transversal trees.


Separable permutations

σ =(

1 2 3 4 5 6 76 7 4 5 1 3 2

)= ((�⊕�)(�⊕�))(�⊕(��)).


Nonseparable permutation - example


Separable tree

DefinitionA positive generic rooted transversal tree is separable if itsassociated permutation is separable.


Passing to the univariate case

QuestionGiven a separable snake σ, is it possible to construct a Morsepolynomial Q : R→ R that realises σ?


Example

=

(�⊕(��)

)⊕ (��) =

(1 2 3 4 51 3 2 5 4

).


The contact tree

a1(x) = 0,a2(x) = x

2,a3(x) = x

2 + x3,a4(x) = x

1,a5(x) = x

1 + x2.


Answer in the univariate case

Theorem ([Sor19a])

Consider m ∈ N and fix a separable (m + 1)-snakeσ : {1, 2, . . . ,m + 1} → {1, 2, . . . ,m + 1} such thatσ(m) > σ(m + 1). Construct the polynomials ai(x) ∈ R[x ]such that their contact tree is one of the binary separatingtrees of σ. Let Qx(y) ∈ R[x ][y ] be

Qx(y) :=

∫ y0

m+1∏i=1

(t − ai(x)

)dt.

Then Qx(y) is a one variable Morse polynomial and forsufficiently small x > 0, the Arnold snake associated to Qx(y)is σ.


Proof

cj(x)− ci(x) = Qx(aj(x)

)− Qx

(ai(x)

)=∫ aj (x)ai (x)

Px(t)dt= (−1)m+1−iSi(x) + . . . + (−1)m+1−jSj(x).

Px(y) :=∏m+1

i=1(y−ai (x))

Qx(y) :=

∫ y0

Px(t)dt

Si(x) :=

∣∣∣∣∣∫ ai+1(x)ai (x)

Px(y)dy

∣∣∣∣∣δ1

δ2

δ3

δ4

a1(x)

a2(x)

a3(x)

a4(x)

a5(x)

S1

S2

S3

S4

y

Px(y)


Propositionνx(Si) = ei +

∑{G∈Gi |G≤CT ai∧ai+1} cG (i)νx(G ).

νx(S4(x)) = 2νx(x1) + 1νx(x2) + 4νx(x3) + νx(x3) = 19.

x1

x6

a1 a2

x2

a3 x3

a4 x4

a5 x5

a6 a7

S1 S2 S3 S4 S5 S6


cj(x)− ci (x) = (−1)m+1−iSi (x) + . . .+ (−1)m+1−jSj(x).

PropositionAmong the valuations νx(Si), νx(Si+1), . . . , νx(Sj−1) theminimum is attained only one time by Si∧j .

Main idea :

cj − ci > 0 ⇔ σ(j) > σ(i).


a1(x) = 0,a2(x) = x

2,a3(x) = x

2 + x3,a4(x) = x

1,a5(x) = x

1 + x2.

Qx(y) :=

∫ y0

5∏i=1

(t − ai(x)

)dt.


Construction of the desired bivariate polynomial f

Theorem ([Sor18])Let σ be a separable (m + 1)-snake, with m an even integer,σ(m) > σ(m + 1). Let f ∈ R[x , y ] be constructed as follows:(a) construct Qx(y) ∈ R[x ][y ],

Qx(y) :=

∫ y0

m+1∏i=1

(t − ai(x)

)dt,

by choosing the polynomials ai(x) ∈ R[x ] such that theircontact tree is one of the binary separating trees of σ.

(b) take f (x , y) := x2 + Qx(y).Then f has a strict local minimum at the origin and thepositive asymptotic snake of f is the given σ.


Proof - strict local minimum

Newton polygon criterion :

If there exists a branch of (f = 0), then f|[AB](a, b) = 0, i.e.

a2 +1

m + 2bm+2 = 0.

Contradiction.


Properties of f

f (x , y) := x2 +

∫ y0

m+1∏i=1

(t − ai(x)

)dt.

- Its positive genericasymptotic Poincaré-Reeb tree:

- It has a strict local minimumat the origin:


Positive-negative contact trees (one variable)1

Pairwise distinct polynomials ai(x) ∈ R[x ] that pass through acommon zero at the origin

1É. Ghys - A singular mathematical promenade, 2017Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 38 / 42

Algorithm flip-flop


Summing-up

f (x , y) := x2 +∫ y

0

∏5i=1

(t − ai(x)

)dt.


Thank you for your attention!


Bibliography:

Miruna-Ştefana Sorea. “The shapes of level curves ofreal polynomials near strict local minima”. PhD thesis.Université de Lille, 2018. url: https://hal.archives-ouvertes.fr/tel-01909028v1(cit. on pp. 11, 12, 20, 22, 23, 45).

Miruna-Stefana Sorea. Constructing Separable ArnoldSnakes of Morse Polynomials. 2019. arXiv:1904.04904 [math.AG] (cit. on p. 36).

Miruna-Stefana Sorea. Measuring the localnon-convexity of real algebraic curves. 2019. arXiv:1907.08585 [math.AG] (cit. on pp. 8, 9).

Miruna-Stefana Sorea. Permutations encoding thelocal shape of level curves of real polynomials viageneric projections. 2019. arXiv: 1910.12790[math.AG] (cit. on p. 15).


https://hal.archives-ouvertes.fr/tel-01909028v1https://hal.archives-ouvertes.fr/tel-01909028v1http://arxiv.org/abs/1904.04904http://arxiv.org/abs/1907.08585http://arxiv.org/abs/1910.12790http://arxiv.org/abs/1910.12790

GoalsPart OneTools

Part Two - Combinatorial interpretations, for x>01. Positive asymptotic snake2. Arnold's snake

Part Three - The construction of positive separable generic rooted transversal treesToolsThe family of polynomials

Part Four - An algorithmSumming-up

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The shapes of level curves of real polynomials near strict local … shapes... · WhenevertheoriginisaMorse strictlocalminimumthe small enough levelcurvesareboundariesof convex topologicaldisks