Representation Theorem for Multivariable Totally Symmetric Functions

TL;DR

This paper proves a representation theorem for multivariable totally symmetric functions, showing they can be expressed via generators of multisymmetric polynomials, and explores the geometric properties and regularity issues of these generators.

Contribution

It introduces a new representation theorem for multisymmetric functions and analyzes the singularity and geometric aspects of the polynomial generators.

Findings

Multisymmetric continuous functions can be decomposed into compositions with generators.

The regularity of the functions may deteriorate after decomposition.

The study provides insights into the singularity and geometric structure of the generators.

Abstract

In this work, we establish a representation theorem for multivariable totally symmetric functions: a multisymmetric continuous function must be the composition of a continuous function and a set of generators of the multisymmetric polynomials. We then study the singularity and geometry of the generators, and show that the regularity may become worse after applying the decomposition.