The mixed degree of families of lattice polytopes

TL;DR

This paper introduces the concept of mixed degree for families of lattice polytopes, generalizing the classical degree, and discusses its properties and relation to mixed volume and existing classifications.

Contribution

It proposes a new notion of mixed degree that extends previous definitions and connects to recent classification results in Ehrhart theory.

Findings

Reformulation of Esterov and Gusev's classification using mixed degree

Introduction of the mixed degree concept generalizing the degree

Relation of mixed degree to mixed volume and previous work

Abstract

The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over the previous years. It is well-known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result of families of $n$ lattice polytopes in $\mathbb{R}^n$ whose mixed volume equals one. Here, we give a reformulation of their result involving the novel notion of a mixed degree that generalizes the degree similar to how the mixed volume generalizes the volume. We discuss and motivate this terminology, and explain why it extends a previous definition of Soprunov. We also remark how a recent combinatorial result due to Bihan solves a related problem posed by Soprunov.