The symmetric function theorem via the Faà di Bruno formula
arXiv:2207.11241
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 Faà di Bruno formula. In two variables, this allows us to completely determine all coefficients that occur in the inductive equations.
Added references, small adjustments. Comments welcome!