# Newton polynomials

Consider the family of symmetric polynomials $\sum^n_{i=1} x_i^k\in\mathbf{Z}[x_1,\ldots,x_n]$. By the fundamental theorem on symmetric polynomials there is a unique Newton poylnomial $N_k\in\mathbf{Z}[x_1,\ldots,x_n]$ such that $\sum^n_{i=1} x_i^k=N_k(s_1,\ldots,s_n)$ with $s_i$ the elementary symmetric polynomials. Is there a way to compute the polynomials $N_k$ by means of e.g. a recursion formula? Thanks!

• Have you looked at Newton's identities? – Jyrki Lahtonen Mar 2 '15 at 18:51
• @JyrkiLahtonen thanks, this is exactly what I was looking for – user220467 Mar 2 '15 at 20:28