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The shapes of level curves of real polynomials near strict local minima Miruna-tefana Sorea Max Planck Institute for Mathematics in the Sciences, Leipzig Algebraic and combinatorial perspectives in the mathematical sciences (ACPMS)

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The shapes of level curves of real polynomials 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

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Goals

  • objects: polynomial

functions f : R2 → R, f (0, 0) = 0 such that O is a strict local minimum;

  • goal: study the real

Milnor fibres of the polynomial (i.e. the level curves (f (x, y) = ε), for 0 < ε ≪ 1, in a small enough neighbourhood of the origin). f (x, y) = x2 + y 2

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 2 / 42

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Whenever the origin is a Morse strict local minimum the small enough level curves are boundaries of convex topological disks.

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Question (Giroux asked Popescu-Pampu, 2004)

Are the small enough level curves of f near strict local minima always boundaries of convex disks? Counterexample by M. Coste: f (x, y) = x2 + (y 2 − x)2.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 4 / 42

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  • Problem: understand these phenomena of non-convexity.
  • Subproblem: construct non-Morse strict local minima

whose nearby small levels are far from being convex.

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Question

What combinatorial object can encode the shape by measuring the non-convexity of a smooth and compact connected component of an algebraic curve in R2?

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The Poincaré-Reeb graph

associated to a curve and to a direction x

Definition

Two points of D are equivalent if they belong to the same connected component of a fibre of the projection Π : R2 → R, Π(x, y) := x.

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The Poincaré-Reeb tree

Theorem ([Sor19b])

The Poincaré-Reeb graph is a transversal tree: it is a plane tree whose open edges are transverse to the foliation induced by the function x; its vertices are endowed with a total preorder relation induced by the function x.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 8 / 42

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The asymptotic Poincaré-Reeb tree

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

Theorem ([Sor19b])

The asymptotic Poincaré-Reeb tree stabilises. It is a rooted tree; the total preorder relation

  • n its vertices is strictly

monotone on each geodesic starting from the root. Impossible asymptotic configuration:

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  • Characterise all possible

topological types of asymptotic Poincaré-Reeb trees.

  • Construct a family of

polynomials realising a large class of transversal trees as their Poincaré-Reeb trees.

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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-Reeb

trees.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 11 / 42

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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-Reeb

trees.

Theorem ([Sor18])

Given any separable positive generic rooted transversal tree, we construct the equation of a real bivariate polynomial with isolated minimum at the origin which realises the given tree as a Poincaré-Reeb tree.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 11 / 42

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Tool 1 : The polar curve

Γ(f , x) :=

  • (x, y) ∈ R2
  • ∂f

∂y (x, y) = 0

  • It is the set of points where the

tangent to a level curve is vertical.

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

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Tool 2 : Choosing a generic projection

Avoid vertical inflections: Avoid vertical bitangents:

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The generic asymptotic Poincaré-Reeb tree

Theorem ([Sor19c])

In the asymptotic case, if the direction x is generic, then we have a total order relation and a complete binary tree. Two inequivalent trees

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Tool 3: The discriminant locus

Φ : R2

x,y → R2 x,z, Φ(x, y) =

  • x, f (x, y)
  • .

The critical locus

  • f Φ is the polar

curve Γ(f , x). The discriminant locus of Φ is the critical image ∆ = Φ(Γ).

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Genericity hypotheses

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

  • the curve Γ+ is reduced;
  • the map Φ|Γ+ : Γ+ → ∆+ is

a homeomorphism.

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  • 1. Positive asymptotic snake

To any positive (i.e. for x > 0) generic asymptotic Poincaré-Reeb tree we can associate a permutation σ, called the positive asymptotic snake.

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  • 2. Arnold’s snake (one variable)

One can associate a permutation to a Morse polynomial, by considering two total order relations on the set of its critical points: Arnold’s snake.

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  • 2. Arnold’s snake (one variable)

The study of asymptotic forms

  • f the graphs of
  • ne variate

polynomials f (x0, y), for x0 tending to zero.

Theorem ([Sor18])

σ = τ.

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Proof

The interplay between the polar curve and the discriminant curve: σ = τ = 1 2 3 2 3 1

  • Miruna-Ştefana Sorea (MPI MiS)

Level curves of real polynomials June 12, 2020 20 / 42

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The construction

Subquestion

Given a generic rooted transversal tree, can we construct the equation of a real bivariate polynomial with isolated minimum at the origin which realises the given tree as a Poincaré-Reeb tree?

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The construction

Subquestion

Given a generic rooted transversal tree, can we construct the equation of a real bivariate polynomial with isolated minimum at the origin which realises the given tree as a Poincaré-Reeb tree?

Theorem ([Sor18])

We give a positive constructive answer: we construct a family of polynomials that realise all separable positive generic rooted transversal trees.

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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 22 / 42

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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 23 / 42

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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 23 / 42

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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 23 / 42

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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 23 / 42

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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 23 / 42

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Separable permutations

σ = 1

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

  • = ((⊡⊕⊡)⊖(⊡⊕⊡))⊖(⊡⊕(⊡⊖⊡)).

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Nonseparable permutation - example

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Separable tree

Definition

A positive generic rooted transversal tree is separable if its associated permutation is separable.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 26 / 42

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Passing to the univariate case

Question

Given a separable snake σ, is it possible to construct a Morse polynomial Q : R → R that realises σ?

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 27 / 42

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Example

=

  • ⊡ ⊕(⊡ ⊖ ⊡)
  • ⊕ (⊡ ⊖ ⊡) =
  • 1 2 3 4 5

1 3 2 5 4

  • .

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 28 / 42

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The contact tree

a1(x) = 0, a2(x) = x2, a3(x) = x2 + x3, a4(x) = x1, a5(x) = x1 + x2.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 29 / 42

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Answer in the univariate case

Theorem ([Sor19a])

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

m+1

  • i=1
  • t − ai(x)
  • dt.

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

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 30 / 42

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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) := y 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 Px(y)

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 31 / 42

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Proposition

νx(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

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

Proposition

Among the valuations νx(Si), νx(Si+1), . . . , νx(Sj−1) the minimum is attained only one time by Si∧j. Main idea : cj − ci > 0 ⇔ σ(j) > σ(i).

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a1(x) = 0, a2(x) = x2, a3(x) = x2 + x3, a4(x) = x1, a5(x) = x1 + x2.

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a1(x) = 0, a2(x) = x2, a3(x) = x2 + x3, a4(x) = x1, a5(x) = x1 + x2. Qx(y) := y

5

  • i=1
  • t − ai(x)
  • dt.

Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 34 / 42

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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) := y

m+1

  • i=1
  • t − ai(x)
  • dt,

by choosing the polynomials ai(x) ∈ R[x] such that their contact 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 the positive asymptotic snake of f is the given σ.

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Proof - strict local minimum

Newton polygon criterion :

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Proof - strict local minimum

Newton polygon criterion :

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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.

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Properties of f

f (x, y) := x2 + y

m+1

  • i=1
  • t − ai(x)
  • dt.
  • Its positive generic

asymptotic Poincaré-Reeb tree:

  • It has a strict local minimum

at the origin:

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Positive-negative contact trees (one variable)1

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

1É. Ghys - A singular mathematical promenade, 2017

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Algorithm flip-flop

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Summing-up

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Summing-up

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Summing-up

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Summing-up

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Summing-up

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Summing-up

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Summing-up

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Summing-up

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Summing-up

f (x, y) := x2 + y 5

i=1

  • t − ai(x)
  • dt.

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Summing-up

f (x, y) := x2 + y 5

i=1

  • t − ai(x)
  • dt.

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Thank you for your attention!

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Bibliography:

Miruna-Ştefana Sorea. “The shapes of level curves of real 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 Arnold Snakes of Morse Polynomials. 2019. arXiv: 1904.04904 [math.AG] (cit. on p. 36). Miruna-Stefana Sorea. Measuring the local non-convexity of real algebraic curves. 2019. arXiv: 1907.08585 [math.AG] (cit. on pp. 8, 9). Miruna-Stefana Sorea. Permutations encoding the local shape of level curves of real polynomials via generic projections. 2019. arXiv: 1910.12790 [math.AG] (cit. on p. 15).

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