Consider a general polynomial non-linear system of equations as follows over the reals:
$$ \begin{array}{c} f_1(x_1, \cdots, x_D) = 0 \\ \vdots \\ f_N(x_1, \cdots, x_D) = 0 \\ \end{array} $$
note that each $f_i$ is a (multivariate) polynomial of finite degree.
I was interested in characterizing and/or finding solutions to such system. I was interested in:
- Determining (possibly in terms of $N$ and $D$ overdetermined and underdetermined systems) when the system had solutions and how many or if at all. Maybe there is more than just looking at those numbers like the specific polynomials that appeared etc.
- Then once we know if it has a solutions, what are the algorithms out there that compute such solutions? I assume there would be many solutions most of the time but I would expect the algorithm to say how to find the solutions its designed to find.
I assume I am not the first one to try to solve such a non-linear polynomial system. Thus, I decided to find out what is out there first. What do we know about the solutions to such as system?