# Reference for a real algebraic geometry problem

Disclaimer: I am not a mathematician by training.

I encountered the following problem in my research. Assume that I have $$N$$ real variables $$x_1, x_2, \dots, x_N$$. I am given $$N$$ homogeneous polynomials in the $$x_i$$ unknowns, each with a different degree. More specifically:

\begin{aligned} P_1 &= \sum_i x_i - c_1\\ P_2 &= \sum_i x_i^2 - c_2\\ &\qquad\vdots \\ P_N &= \sum_i x_i^N - c_N \end{aligned}

where $$c_1, c_2, \dots, c_N$$ are given real coefficients. I need to find, if they exist, real solutions of the above equations.

I am asking for references where I can learn the tools needed to attack this type of problems.

Thank you.

There are Newton's formulas expressing elementary symmetric polynomials in terms of power sums. Since your equations are $$P_k=c_k$$, where $$P_k$$ are power sums, you can express the elementary symmetric polynomials $$A_k$$ in terms of your $$c_k$$. Then your solutions $$(x_1,\ldots,x_N)$$ are permutations of the set of roots of the polynomial
$$x^N+A_1x^{N-1}+\cdots+A_N.$$
• Thank you for your answer. Would you have any comments on the following generalization of the problem: Instead of the equations above, we have: \begin{aligned} P_1 &= \sum_i \lambda_i x_i - c_1\\ P_2 &= \sum_i \lambda_i x_i^2 - c_2\\ &\qquad\vdots \\ P_N &= \sum_i \lambda_i x_i^N - c_N \end{aligned} where again $\lambda_i$ are real and satisfy the condition $\sum_i \lambda_i = c_0$. – SymGen Jul 18 '20 at 2:10