Multisymmetric polynomials in dimension three

TL;DR

This paper investigates the algebraic structure of multisymmetric polynomials in three dimensions, identifying key relations that generate the ideal of relations among minimal generators, with implications for higher dimensions.

Contribution

It introduces new relations of degree five and six that minimally generate the ideal of relations in three dimensions and presents a degree $2n$ relation for arbitrary dimensions.

Findings

Relations of degree five and six generate the ideal in 3D.

A new degree $2n$ relation among polarized power sums is identified.

The results extend to arbitrary dimensions with specific relations.

Abstract

The polarizations of one relation of degree five and two relations of degree six minimally generate the ideal of relations among a minimal generating system of the algebra of multisymmetric polynomials in an arbitrary number of three-dimensional vector variables. In the general case of $n$-dimensional vector variables, a relation of degree $2n$ among the polarized power sums is presented such that it is not contained in the ideal generated by lower degree relations.