On the difference between `tropical functions' and real-valued functions

TL;DR

This paper explores the concept of integral closure in idempotent semirings, linking tropical functions to real functions through convex geometry and algebraic structures.

Contribution

It introduces integral closure for elements and ideals in idempotent semirings and relates it to convex bodies, clarifying the difference between tropical and real functions.

Findings

Integral closure corresponds to its algebraic counterpart in commutative algebra.

In free semirings, integral closure relates to convex bodies under Minkowski sum.

The concept explains the fundamental difference between tropical functions and real-valued functions.

Abstract

I introduce the concept of integral closure for elements and ideals in idempotent semirings, and establish how it corresponds to its namesake in commutative algebra. In the case of free semirings, integral closure can be understood in terms of a certain monoid of convex bodies under Minkowski sum. I argue that integral closure therefore accounts for the difference between `tropical functions' and real functions.