Separating invariants for multisymmetric polynomials

TL;DR

This paper advances the understanding of separating invariants for multisymmetric polynomials by reducing the number of variable sets needed and providing explicit minimal sets for small n in characteristic zero or large characteristic fields.

Contribution

It proves that separating invariants depend only on a limited number of variable sets and explicitly constructs minimal separating sets for small n in certain fields.

Findings

Reducing the variable sets needed for separating invariants to 6; 6; 6; + 1 sets.

Explicit minimal separating sets for n  4 in characteristic 0 or > n fields.

Improves a previous general result by Domokos.

Abstract

This article studies separating invariants for the ring of multisymmetric polynomials in $m$ sets of $n$ variables over an arbitrary field $\mathbb{K}$. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on $\lfloor \frac{n}{2} \rfloor + 1$ sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for $n \leq 4$ we explicitly give minimal separating sets (with respect to inclusion) for all $m$ in case $\text{char}(\mathbb{K}) = 0$ or $\text{char}(\mathbb{K}) > n$.