the shapes of level curves of real polynomials near strict local … shapes... ·...
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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
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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
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 thesmall enough level curves are boundaries of convex
topological disks.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 3 / 42
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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.
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 minimawhose nearby small levels are far from being convex.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 5 / 42
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Question
What combinatorial object can encode the shape bymeasuring the non-convexity of a smooth and compactconnected component of an algebraic curve in R2?
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 6 / 42
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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 .
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 7 / 42
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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 .
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-Reebtree stabilises. It is a rootedtree; the total preorder relationon its vertices is strictlymonotone on each geodesicstarting from the root.
Impossible asymptoticconfiguration:
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 9 / 42
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• Characterise all possibletopological types ofasymptotic Poincaré-Reebtrees.
• Construct a family ofpolynomials realising alarge class of transversaltrees as theirPoincaré-Reeb trees.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 10 / 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-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.
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-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.
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 thetangent to a level curve isvertical.
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:
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 13 / 42
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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
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 14 / 42
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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).
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 15 / 42
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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.
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 17 / 42
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2. Arnold’s snake (one variable)
One can associatea permutation to aMorsepolynomial, byconsidering twototal orderrelations on theset of its criticalpoints: Arnold’ssnake.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 18 / 42
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2. Arnold’s snake (one variable)
The study ofasymptotic formsof the graphs ofone variatepolynomialsf (x0, y), for x0tending to zero.
Theorem([Sor18])
σ = τ.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 19 / 42
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ProofThe interplay between the polar curve and the discriminantcurve:
σ = τ =
(1 2 32 3 1
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 21 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 21 / 42
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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 76 7 4 5 1 3 2
)= ((�⊕�)(�⊕�))(�⊕(��)).
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 24 / 42
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Nonseparable permutation - example
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 25 / 42
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Separable tree
DefinitionA positive generic rooted transversal tree is separable if itsassociated 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
QuestionGiven a separable snake σ, is it possible to construct a Morsepolynomial 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 51 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) = x
2,a3(x) = x
2 + x3,a4(x) = x
1,a5(x) = x
1 + 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)-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.
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) :=
∫ 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)
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 31 / 42
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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
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 32 / 42
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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).
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 34 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 34 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 34 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 34 / 42
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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.
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) :=
∫ 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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 35 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 36 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 36 / 42
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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.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 36 / 42
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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:
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 37 / 42
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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
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Algorithm flip-flop
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 39 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials June 12, 2020 40 / 42
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=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 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).
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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