The symmetric function theorem via the Fa\`a di Bruno formula

TL;DR

This paper provides a new proof of the symmetric function theorem using the multivariate Faà di Bruno formula, connecting polynomial invariance under permutation to elementary symmetric polynomials.

Contribution

It introduces an analytic approach to the symmetric function theorem by leveraging the multivariate Faà di Bruno formula, especially in two variables.

Findings

Complete determination of coefficients in two-variable inductive equations.

New analytic proof of the symmetric function theorem.

Enhanced understanding of polynomial invariance in multivariate calculus.

Abstract

The symmetric function theorem states that a polynomial that is invariant under permutation of variables, is a polynomial in the elementary symmetric polynomials. We deduce this classical result, in the analytic setting, from the multivariate Fa\`a di Bruno formula. In two variables, this allows us to completely determine all coefficients that occur in the inductive equations.