Classification of Triples of Lattice Polytopes with a Given Mixed Volume

Gennadiy Averkov; Christopher Borger; Ivan Soprunov

doi:10.1007/s00454-020-00246-4

Classification of Triples of Lattice Polytopes with a Given Mixed Volume

Gennadiy Averkov, Christopher Borger, Ivan Soprunov

Research output: Contribution to journal › Article › peer-review

Abstract

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volumemin dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

Original language	American English
Journal	Discrete Computational Geometry
DOIs	https://doi.org/10.1007/s00454-020-00246-4
State	Published - Oct 1 2020

Keywords

Bernstein-Khovanskii-Kouchnirenko theorem; Classification; Lattice polytope; Mixed volume; Newton polytope; Sparse polynomial systems

Disciplines

Mathematics

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10.1007/s00454-020-00246-4License: CC BY

Classification of Triples of Lattice Polytopes with a Given MixedFinal published version, 1.23 MBLicense: CC BY

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title = "Classification of Triples of Lattice Polytopes with a Given Mixed Volume",

abstract = " We present an algorithm for the classification of triples of lattice polytopes with a given mixed volumemin dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.",

keywords = "Bernstein-Khovanskii-Kouchnirenko theorem; Classification; Lattice polytope; Mixed volume; Newton polytope; Sparse polynomial systems",

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AU - Averkov, Gennadiy

AU - Borger, Christopher

AU - Soprunov, Ivan

PY - 2020/10/1

Y1 - 2020/10/1

N2 - We present an algorithm for the classification of triples of lattice polytopes with a given mixed volumemin dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

AB - We present an algorithm for the classification of triples of lattice polytopes with a given mixed volumemin dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed m. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this leads to a description of all generic trivariate sparse polynomial systems that are solvable by radicals.

KW - Bernstein-Khovanskii-Kouchnirenko theorem; Classification; Lattice polytope; Mixed volume; Newton polytope; Sparse polynomial systems

UR - https://engagedscholarship.csuohio.edu/scimath_facpub/338

U2 - 10.1007/s00454-020-00246-4

DO - 10.1007/s00454-020-00246-4

M3 - Article

JO - Discrete Computational Geometry

JF - Discrete Computational Geometry

ER -

Classification of Triples of Lattice Polytopes with a Given Mixed Volume

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