# Decomposing tropical rational functions

## TL;DR

This paper presents algorithms for decomposing tropical univariate rational functions into compositions of simpler tropical binomials and trinomials, with special cases for monotone functions and criteria for commutativity of compositions.

## Contribution

It introduces novel algorithms for tropical function decomposition and provides criteria for the commutativity of tropical polynomial compositions.

## Key findings

- Algorithms successfully decompose tropical rational functions.
- Monotone functions decompose into binomials only.
- Criteria established for when tropical polynomial compositions commute.

## Abstract

An algorithm is designed which decomposes a tropical univariate rational function into a composition of tropical binomials and trinomials. When a function is monotone, the composition consists just of binomials. Similar algorithms are designed for decomposing tropical algebraic rational functions being (in the classical language) piece-wise linear functions with rational slopes of their linear pieces. In addition, we provide a criterion when the composition of two tropical polynomials commutes (for classical polynomials a similar question was answered by J.~Ritt).