Constructing polynomial systems with many positive solutions using tropical geometry

Hilany, Boulos El

doi:10.1007/s13163-017-0254-1

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…[4,24,20], and the references therein). Some of the exposition and notations here are taken from [5,3,7]. We start by introducing in § 2.1 the non-trivially-valued field K over which the polynomial maps will be defined.…”

Section: Tropical Geometry and Polynomial Mapsmentioning

confidence: 99%

The tropical discriminant of a polynomial map on a plane

Hilany¹

2022

Preprint

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The discriminant of a polynomial map between two spaces is an object central to many problems in mathematics. Standard methods for computing it rely on elimination techniques which can be inefficient. This paper concerns polynomial maps on the two-dimensional torus defined over the field of Puiseux series with complex coefficients. We present a combinatorial procedure for computing the tropical curve of the discriminant of such maps that satisfy some mild genericity conditions. Thanks to the advances made in tropical geometry, our results give rise to a new tool for investigating the discriminant of a complex polynomial map on the plane, and for computing its Newton polytope.

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Section: Tropical Geometry and Polynomial Mapsmentioning

confidence: 99%

The tropical discriminant of a polynomial map on a plane

Hilany¹

2022

Preprint

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The Tropical Non-Properness Set of a Polynomial Map

El Hilany

2024

Discrete Comput Geom

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We study some discrete invariants of Newton non-degenerate polynomial maps $$f: {\mathbb {K}}^n \rightarrow {\mathbb {K}}^n$$ f : K n → K n defined over an algebraically closed field of Puiseux series $${\mathbb {K}}$$ K , equipped with a non-trivial valuation. It is known that the set $${\mathcal {S}}(f)$$ S ( f ) of points at which f is not finite forms an algebraic hypersurface in $${\mathbb {K}}^n$$ K n . The coordinate-wise valuation of $${\mathcal {S}}(f)\cap ({\mathbb {K}}^*)^n$$ S ( f ) ∩ ( K ∗ ) n is a piecewise-linear object in $${\mathbb {R}}^n$$ R n , which we call the tropical non-properness set of f. We show that the tropical polynomial map corresponding to f has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of f. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of $${\mathcal {S}}(f)$$ S ( f ) in terms of multivariate resultants.

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A Polyhedral Method for Sparse Systems with Many Positive Solutions

Bihan¹,

Santos²,

Spaenlehauer³

2018

SIAM J. Appl. Algebra Geometry

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We investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest positively decorable subcomplex of the triangulation. Here, positive decorability is a property that we introduce and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produce large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we get new lower bounds, some of them being the best currently known, for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also study the asymptotics of these numbers and observe a log-concavity property.

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Constructing polynomial systems with many positive solutions using tropical geometry

Cited by 3 publications

References 18 publications

The tropical discriminant of a polynomial map on a plane

The tropical discriminant of a polynomial map on a plane

The Tropical Non-Properness Set of a Polynomial Map

A Polyhedral Method for Sparse Systems with Many Positive Solutions

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