Classification of triples of lattice polytopes with a given mixed volume

TL;DR

This paper introduces an algorithm to classify triples of lattice polytopes with a fixed mixed volume in three dimensions, enabling the enumeration of all such irreducible triples up to volume 4, and characterizing solvable polynomial systems.

Contribution

It provides a novel algorithm for classifying lattice polytope triples with a given mixed volume, extending understanding of sparse polynomial systems and their solvability.

Findings

Enumerated all irreducible triples with mixed volume up to 4

Classified generic trivariate sparse polynomial systems with up to 4 solutions

Connected classification to solvability by radicals for these systems

Abstract

We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume $m$ in 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.