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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!

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

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