Tropical Invariants for Permutation Group Actions

Paper

Abstract

We consider the action of a permutation group

of order

on the tropical polynomial semiring in

variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if

is generated by

-cycles. There do exist finitely many separating invariants of degree at most

. Separating tropical invariants can be used to construct bi-Lipschitz embeddings of the orbit space

into Euclidean space. We also show that the invariant polynomials of degree

generate the semifield of invariant rational tropical functions, where

are the first

prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative.