Suppose we have n variables $x_1, x_2, ..., x_n$, and denote the $p^{th}$ power sum of these variables $S_p$
$S_p(x_1, ..., x_n) = \sum_{i=1}^n x_i^p$
Is there a method to compute (or approximate) this power sum without using elementary symmetric polynomials (Girard-Newton formulae) or similar methods that require computing individual products of different $x_i$'s?
That is a method such as:
$S_p(x_1, ..., x_n) = \sum_{i=1}^n x_i^p = f( g(x_1, ..., x_n) , p)$
where one can first compute $g(x_1, ..., x_n)$ as a sum or a product of all numbers, and then apply the exponent $p$ or some function of it (probably sounds like Lander, Parkin, and Selfridge conjecture).
I tried log identities (e.g., $log_b(\prod x_i) = \sum log_b x_i$) but it didn't work, obviously.
An approximation rather than an exact method would be welcomed too.