How to Tropicalize a non-Archimedean Lattice

TL;DR

This paper introduces a novel method for computing the tropicalization of lattices over valuation rings, utilizing entropy polynomials and tropicalization of Haar measures, advancing tropical geometry techniques.

Contribution

It presents a new computational approach for tropicalizing lattices over valuation rings, linking tropical geometry with measure theory.

Findings

Developed a method to compute tropicalizations of lattices.

Connected tropicalization with Haar measures on local fields.

Constructed tropical semimodules supported by polyhedral complexes.

Abstract

The tropicalization of a linear space over a non-archimedean field is a tropical linear space. In this paper, we present a method for computing the tropicalization of any lattice over a valuation ring. The resulting tropical semimodule is the support of a polyhedral complex constructed from a certain multilinear polynomial we call the entropy polynomial. The key idea in our argument is the tropicalization of Haar measures on lattices over local fields.