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!
2026-03-29 19:53:34.1774814014
Newton polynomials
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As far as I remember there is this :
http://www.imm.dtu.dk/arith21/presentations/pres_92.pdf
I assume thay you will want to evalute them at points.
(Newton's identites are theoretically good, but practically intractable when the number of variables becomes big.)