Tropical Invariants for Permutation Group Actions

We consider the action of a permutation group $G$ of order $k$ on the tropical polynomial semiring in $n$ variables. We prove that the sub-semiring of invariant polynomials is finitely generated if and only if $G$ is generated by $2$-cycles. There do exist finitely many separating invariants of degree at most $\max \{n,\left(\genfrac{}{}{0px}{}{n}{2}\right)\}$. Separating tropical invariants can be used to construct bi-Lipschitz embeddings of the orbit space ${\mathbb{R}}^n\mathrm{/}G$ into Euclidean space. We also show that the invariant polynomials of degree $\le n{p}_1{p}_2\cdots p_k$ generate the semifield of invariant rational tropical functions, where $p_1,p_2,\dots ,p_k$ are the first $k$ prime numbers. Most results are also true over arbitrary semirings that are additively idempotent and multiplicatively cancellative.