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Expressive curves Sergey Fomin University of Michigan arXiv:2006.14066 (with E. Shustin) arXiv:1711.10598 (with P. Pylyavskyy, E. Shustin, D. Thurston) Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 1 / 28 Rolles

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Expressive curves

Sergey Fomin

University of Michigan

arXiv:2006.14066 (with E. Shustin) arXiv:1711.10598 (with P. Pylyavskyy, E. Shustin, D. Thurston)

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 1 / 28

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Rolle’s Theorem

Theorem

Let g(x) ∈ R[x] be a polynomial whose roots are real and distinct. Then g has exactly one critical point between each pair of consecutive roots, and no other critical points (even over C).

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Expressive curves

G(x, y) ∈ R[x, y] ⊂ C[x, y] polynomial with real coefficients C = {(x, y) ∈ C2 | G(x, y) = 0} affine plane algebraic curve CR ={(x, y) ∈ R2 | G(x, y) = 0} set of real points of C

Definition

Polynomial G (resp., curve C) is called expressive if

all critical points of G are real;
at each critical point, G has a nondegenerate Hessian;
each bounded connected component of R2\CR contains exactly
ne critical point of G;
each unbounded component of R2\CR contains no critical points;
CR is connected, and contains infinitely many points.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 3 / 28

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Expressive curves

G(x, y) ∈ R[x, y] ⊂ C[x, y] polynomial with real coefficients C = {(x, y) ∈ C2 | G(x, y) = 0} affine plane algebraic curve CR ={(x, y) ∈ R2 | G(x, y) = 0} set of real points of C

Definition

Polynomial G (resp., curve C) is called expressive if

all critical points of G are real;
at each critical point, G has a nondegenerate Hessian;
each bounded connected component of R2\CR contains exactly
ne critical point of G;
each unbounded component of R2\CR contains no critical points;
CR is connected, and contains infinitely many points.

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Example of an expressive curve

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Example of a non-expressive curve

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Motivations

Our main result is a complete classification of expressive curves (subject to a mild technical condition). Why care?

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Motivations

Our main result is a complete classification of expressive curves (subject to a mild technical condition). Why care?

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 6 / 28

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Motivations

Our main result is a complete classification of expressive curves (subject to a technical condition). Why care? Motivation #1: Extending the theory of hyperplane arrangements

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From plane curves to cluster theory

Motivation #2: Understanding the geometry and topology of plane curves using combinatorics of quiver mutations and plabic graphs

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Curve → divide

A nodal curve in the real affine plane defines a divide. There is a local version of this construction, involving morsifications.

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Curve → divide

A nodal curve in the real affine plane defines a divide. There is a local version of this construction, involving morsifications.

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Divide → plabic graph

Plabic (planar bicolored) graphs were introduced by A. Postnikov to study parametrizations of cells in totally nonnegative Grassmannians. All our plabic graphs are trivalent-univalent. Any divide gives rise to a plabic graph: − → − →

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Divide → plabic graph

Plabic (planar bicolored) graphs were introduced by A. Postnikov to study parametrizations of cells in totally nonnegative Grassmannians. All our plabic graphs are trivalent-univalent. Any divide gives rise to a plabic graph: − → − →

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 10 / 28

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Divide → plabic graph

Plabic (planar bicolored) graphs were introduced by A. Postnikov to study parametrizations of cells in totally nonnegative Grassmannians. All our plabic graphs are trivalent-univalent. Any divide gives rise to a plabic graph: − → − →

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 10 / 28

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Move equivalence of plabic graphs

Two plabic graphs are called move equivalent if they can be obtained from each other via repeated application of the following moves: flip moves ← → ← → square move ← →

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Plabic graph → quiver

Any plabic graph defines a quiver: Square moves on plabic graphs translate into quiver mutations: Flip moves do not change the quiver.

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Plabic graph → quiver

Any plabic graph defines a quiver: Square moves on plabic graphs translate into quiver mutations: Flip moves do not change the quiver.

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Curve → divide → plabic graph → quiver

Conjecture

Two plabic graphs coming from expressive curves are move equivalent if and only if their quivers are mutation equivalent.

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Curve → divide → plabic graph → quiver

Conjecture

Two plabic graphs coming from expressive curves are move equivalent if and only if their quivers are mutation equivalent.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 13 / 28

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Curve → divide → plabic graph → link

There is a construction [T. Kawamura + FPST] that associates a canonical (transverse) link to any plabic graph.

Theorem (SF-P. Pylyavskyy-E. Shustin-D. Thurston)

The link of a plabic graph is invariant under local moves.

Theorem (N. A’Campo + FPST)

The link of a divide arising from a real morsification of a plane curve singularity is isotopic to the link of the singularity. We conjecture that under mild technical assumptions, the link of a divide arising from an expressive curve is isotopic to the curve’s link at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 14 / 28

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Curve → divide → plabic graph → link

There is a construction [T. Kawamura + FPST] that associates a canonical (transverse) link to any plabic graph.

Theorem (SF-P. Pylyavskyy-E. Shustin-D. Thurston)

The link of a plabic graph is invariant under local moves.

Theorem (N. A’Campo + FPST)

The link of a divide arising from a real morsification of a plane curve singularity is isotopic to the link of the singularity. We conjecture that under mild technical assumptions, the link of a divide arising from an expressive curve is isotopic to the curve’s link at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 14 / 28

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Curve → divide → plabic graph → link

There is a construction [T. Kawamura + FPST] that associates a canonical (transverse) link to any plabic graph.

Theorem (SF-P. Pylyavskyy-E. Shustin-D. Thurston)

The link of a plabic graph is invariant under local moves.

Theorem (N. A’Campo + FPST)

The link of a divide arising from a real morsification of a plane curve singularity is isotopic to the link of the singularity. We conjecture that under mild technical assumptions, the link of a divide arising from an expressive curve is isotopic to the curve’s link at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 14 / 28

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Curve → divide → plabic graph → link

There is a construction [T. Kawamura + FPST] that associates a canonical (transverse) link to any plabic graph.

Theorem (SF-P. Pylyavskyy-E. Shustin-D. Thurston)

The link of a plabic graph is invariant under local moves.

Theorem (N. A’Campo + FPST)

The link of a divide arising from a real morsification of a plane curve singularity is isotopic to the link of the singularity. We conjecture that under mild technical assumptions, the link of a divide arising from an expressive curve is isotopic to the curve’s link at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 14 / 28

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Mutation equivalence vs. link equivalence

curve germ divide Combinatorics Geometry plabic graph quiver link move equivalence mutation transverse equivalence isotopy

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Mutation equivalence vs. link equivalence

curve germ divide Combinatorics Geometry plabic graph quiver link move equivalence mutation transverse equivalence isotopy

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Back to expressive curves

A real plane algebraic curve C is expressive if its defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of C.

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L∞-regular curves

x, y, z projective coordinates in P2 L∞ = {z = 0} line at infinity C2 = P2\L∞ affine complex plane

Definition

A projective curve C = Z(F) ⊂ P2 is called L∞-regular if ∀p ∈ C ∩ L∞ (Z( ∂F

∂x ) · Z( ∂F ∂y ))p = µ(C, p) + (C · L∞)p − 1.

An affine curve C ⊂ C2 is called L∞-regular if its projective closure

C ⊂ P is L∞-regular.

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L∞-regular curves

x, y, z projective coordinates in P2 L∞ = {z = 0} line at infinity C2 = P2\L∞ affine complex plane

Definition

A projective curve C = Z(F) ⊂ P2 is called L∞-regular if ∀p ∈ C ∩ L∞ (Z( ∂F

∂x ) · Z( ∂F ∂y ))p = µ(C, p) + (C · L∞)p − 1.

An affine curve C ⊂ C2 is called L∞-regular if its projective closure

C ⊂ P is L∞-regular.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 17 / 28

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Polynomial and trigonometric curves

Definition

A real rational curve C ⊂ C2 is called polynomial if it admits a real polynomial parametrization t → (P(t), Q(t)).

Proposition

C is polynomial ⇔ C is a real rational curve with one place at infinity.

Definition

A real rational curve C ⊂C2 is called trigonometric if CR admits a real trigonometric parametrization t → (P(cos t, sin t), Q(cos t, sin t)).

Proposition

C is trigonometric ⇔ C is a real rational curve with an infinite real point set and with two complex conjugate places at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 18 / 28

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Polynomial and trigonometric curves

Definition

A real rational curve C ⊂ C2 is called polynomial if it admits a real polynomial parametrization t → (P(t), Q(t)).

Proposition

C is polynomial ⇔ C is a real rational curve with one place at infinity.

Definition

A real rational curve C ⊂C2 is called trigonometric if CR admits a real trigonometric parametrization t → (P(cos t, sin t), Q(cos t, sin t)).

Proposition

C is trigonometric ⇔ C is a real rational curve with an infinite real point set and with two complex conjugate places at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 18 / 28

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Polynomial and trigonometric curves

Definition

A real rational curve C ⊂ C2 is called polynomial if it admits a real polynomial parametrization t → (P(t), Q(t)).

Proposition

C is polynomial ⇔ C is a real rational curve with one place at infinity.

Definition

A real rational curve C ⊂C2 is called trigonometric if CR admits a real trigonometric parametrization t → (P(cos t, sin t), Q(cos t, sin t)).

Proposition

C is trigonometric ⇔ C is a real rational curve with an infinite real point set and with two complex conjugate places at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 18 / 28

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Polynomial and trigonometric curves

Definition

A real rational curve C ⊂ C2 is called polynomial if it admits a real polynomial parametrization t → (P(t), Q(t)).

Proposition

C is polynomial ⇔ C is a real rational curve with one place at infinity.

Definition

A real rational curve C ⊂C2 is called trigonometric if CR admits a real trigonometric parametrization t → (P(cos t, sin t), Q(cos t, sin t)).

Proposition

C is trigonometric ⇔ C is a real rational curve with an infinite real point set and with two complex conjugate places at infinity.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 18 / 28

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Expressivity criterion

Theorem

Let C ⊂ C2 be a reduced real algebraic curve, with all irreducible components real. The following are equivalent:

C is expressive and L∞-regular;
each component of C is either trigonometric or polynomial,

all singular points of C in the affine plane are hyperbolic nodes, and the set of real points of C in the affine plane is connected. In particular, any polynomial or trigonometric curve all of whose singular points (away from infinity) are real hyperbolic nodes is both expressive and L∞-regular.

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Expressivity criterion

Theorem

Let C ⊂ C2 be a reduced real algebraic curve, with all irreducible components real. The following are equivalent:

C is expressive and L∞-regular;
each component of C is either trigonometric or polynomial,

all singular points of C in the affine plane are hyperbolic nodes, and the set of real points of C in the affine plane is connected. In particular, any polynomial or trigonometric curve all of whose singular points (away from infinity) are real hyperbolic nodes is both expressive and L∞-regular.

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Constructing expressive curves

We describe many procedures for constructing new expressive curves from existing examples.

4y2−3y−x=0 4(x2+y)2−3(x2+y)−x=0 4(x2+y2)2−3(x2+y2)−x=0

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Example I: Line arrangements

A nodal connected real line arrangement is an expressive curve.

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Example II: Arrangements of parabolas

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Example III: Circle arrangements

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Example IV: Arrangements of lines and circles

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Example V: Arrangements of nodal cubics

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Example VI: Lissajous-Chebyshev curves

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Example VII: Hypotrochoids and epitrochoids

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Pseudoline arrangements

A pseudoline arrangement is a connected divide whose branches are embedded intervals, and any two of them intersect at most once.

Proposition

A pseudoline arrangement comes from a morsification of an isolated plane curve singularity iff any two pseudolines in it intersect.

Proposition

A pseudoline arrangement comes from an expressive L∞-regular curve (with all components real) iff it is stretchable. Thus there are divides which come from morsifications but not from expressive curves, or vice versa.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 28 / 28

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Pseudoline arrangements

A pseudoline arrangement is a connected divide whose branches are embedded intervals, and any two of them intersect at most once.

Proposition

A pseudoline arrangement comes from a morsification of an isolated plane curve singularity iff any two pseudolines in it intersect.

Proposition

A pseudoline arrangement comes from an expressive L∞-regular curve (with all components real) iff it is stretchable. Thus there are divides which come from morsifications but not from expressive curves, or vice versa.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 28 / 28

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Pseudoline arrangements

A pseudoline arrangement is a connected divide whose branches are embedded intervals, and any two of them intersect at most once.

Proposition

A pseudoline arrangement comes from a morsification of an isolated plane curve singularity iff any two pseudolines in it intersect.

Proposition

A pseudoline arrangement comes from an expressive L∞-regular curve (with all components real) iff it is stretchable. Thus there are divides which come from morsifications but not from expressive curves, or vice versa.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 28 / 28

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Pseudoline arrangements

A pseudoline arrangement is a connected divide whose branches are embedded intervals, and any two of them intersect at most once.

Proposition

A pseudoline arrangement comes from a morsification of an isolated plane curve singularity iff any two pseudolines in it intersect.

Proposition

A pseudoline arrangement comes from an expressive L∞-regular curve (with all components real) iff it is stretchable. Thus there are divides which come from morsifications but not from expressive curves, or vice versa.

Sergey Fomin (University of Michigan) Expressive curves August 4, 2020 28 / 28