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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)
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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
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Whenever the origin is a Morse strict local minimum thesmall
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 minimaalways boundaries of
convex disks?
Counterexample by M. Coste: f (x , y) = x2 + (y 2 − x)2.
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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.
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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?
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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 .
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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 .
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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:
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• Characterise all possibletopological types ofasymptotic
Poincaré-Reebtrees.
• Construct a family ofpolynomials realising alarge class of
transversaltrees as theirPoincaré-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-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.
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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.
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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.
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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 thedirection x is
generic, then wehave a total order relationand a complete binary
tree.
Two inequivalent trees
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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).
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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.
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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.
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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.
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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.
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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.
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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials
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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials
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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials
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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials
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Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials
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Separable permutations
σ =(
1 2 3 4 5 6 76 7 4 5 1 3 2
)= ((�⊕�)(�⊕�))(�⊕(��)).
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Nonseparable permutation - example
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Separable tree
DefinitionA positive generic rooted transversal tree is
separable if itsassociated permutation is separable.
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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?
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Example
=
(�⊕(��)
)⊕ (��) =
(1 2 3 4 51 3 2 5 4
).
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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.
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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
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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)
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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
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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.
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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.
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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.
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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
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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.
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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.
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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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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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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 +
∫ y0
m+1∏i=1
(t − ai(x)
)dt.
- Its positive genericasymptotic Poincaré-Reeb tree:
- It has a strict local minimumat the origin:
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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 /
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Algorithm flip-flop
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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Summing-up
f (x , y) := x2 +∫ y
0
∏5i=1
(t − ai(x)
)dt.
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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).
Miruna-Ştefana Sorea (MPI MiS) Level curves of real polynomials
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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