THE VOLUME POLYNOMIAL OF LATTICE POLYGONS

Ivan Soprunov; Jenya Soprunova

doi:10.1090/proc/16938

THE VOLUME POLYNOMIAL OF LATTICE POLYGONS

Ivan Soprunov
, Jenya Soprunova

Kent State University

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

Abstract

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

Original language	English
Pages (from-to)	5313-5325
Number of pages	13
Journal	Proceedings of the American Mathematical Society
Volume	152
Issue number	12
DOIs	https://doi.org/10.1090/proc/16938
State	Published - Dec 1 2024

Keywords

integer quadratic form
intersection number
lattice polytope
mixed volume
toric surface
tropical curve
Volume polynomial

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10.1090/proc/16938

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AB - We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

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KW - toric surface

KW - tropical curve

KW - Volume polynomial

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THE VOLUME POLYNOMIAL OF LATTICE POLYGONS

Abstract

Keywords

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