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.