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Suppose I have a set of power sum symmetric polynomial as $$S_p =\sum_i^N x^p_i ~~;~~~~~~~~p=\{1,N\}$$ and I have N of them $\{S_1...S_N\}$ Question is given this, can we find ${x_n=F(\{S_p\})}$?

For N=2 this is possible through direct check, For N=2,

This gives $$S_1 = X_1 + X_2\\ S_2 = X_1^2+X_2^2$$ One can find $X_1X_2 = \frac{S_1^2-S_2^2}{2}$ and then use 1st equation to get $X_1=(-S_1 \pm \sqrt{4S_2^2 - 2S_1^2})/2$

but I want to know if there is a general procedure doing that for any N (which i doubt because of non-uniqueness,but may be one can say something with Newtons laws?) or even for large N limit?

Jyrki Lahtonen
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Ars3nous
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1 Answers1

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By Newton's identities you can find the values of the elementary symmetric functions of the unknowns $X_i$. In other words when we consider the equation $$ (X-X_1)(X-X_2)\cdots(X-X_N)=X^N+a_1X^{N-1}+\cdots +a_N=0 $$ that has the unknown numbers $X_i$ as its solutions, we can calculate coefficients $a_i, i=1,2,\ldots,N,$ in terms of the power sums $S_i, i=1,2,\ldots,N$.

But there is no formula in terms of the usual arithmetic operations and radicals to find the values of the quantities $X_i$ when $N\ge5$. This is because no such formula for the zeros of high degree polynomials can exist.

Non-uniqueness is IMO a non-problem. If you can find the solutions of that equation by some means, the solutions form a permutation of the $X_i$s. You just cannot tell which is which, but this was hopefully clear from the beginning.

Jyrki Lahtonen
  • 140,891