A minimal set of generators for the ring of multisymmetric functions

TL;DR

This paper provides an explicit minimal generating set for the ring of multisymmetric functions over any commutative ring, improving degree bounds and extending known results beyond characteristic zero.

Contribution

It introduces a minimal set of generators for multisymmetric functions over any commutative ring, generalizing classical results and refining degree bounds.

Findings

Explicit minimal generators for multisymmetric functions

Sharp degree bounds on generators

Extension of classical results to arbitrary rings

Abstract

The purpose of this article is to give, for any commutative ring A, an explicit minimal set of generators for the ring of multisymmetric functions TS^d_A(A[x_1,...,x_r]) as an A-algebra. In characteristic zero, i.e. when A is an algebra over the rational numbers, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously obtained by Fleischmann.