Asymptotic Behavior of Tropical Rank Functions

TL;DR

This paper investigates the asymptotic properties of tropical rank functions on curves, introducing tropical volume notions and establishing their compatibility with tropicalization, paralleling algebraic geometry volume theory.

Contribution

It introduces tropical volume concepts for divisors and modules, proving their optimal asymptotic behavior and compatibility with tropicalization of curves.

Findings

Tropical volume notions are introduced for divisors and modules.

Optimal asymptotic results are established for these tropical volumes.

Tropical volume is shown to be compatible with tropicalization of curves.

Abstract

We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and study tropical notions of volume associated to both divisors and tropical modules. We prove optimal asymptotic results for each case. In addition, we show that the tropical volume is compatible with the tropicalization of curves.