Is it possible to know or get an reasonable upper bound on the number of solutions to an $2$D Polynomial equation ($n$D)?

Question

I have two arbitrary but finite order $2$D polynomial functions $P_1(\vec x)$ and $P_2(\vec x)$, both on the domain $[-1,1]^2$. If I construct an algebraic equation $P_1(\vec x)^2 + P_2(\vec x)^2 = 0$, and know that there exists at least one real solution, is there a way to know more about possible other real solutions. Is there some theory of algebra (possibly Galois theory) that allows me to determine if there exist more real solutions? Mind you, I do not care what the real solutions are, I only wish to know if there are more than one. If this is not possible does there exist some way of putting a useful upper bound on the number of real solutions? Additionally, if I had $n$ polynomials is there a way to accomplish the same thing in $n$D. In other words I want to know if $$ \sum_{k=1}^n P_k^2(\vec x) = 0 $$ has only one real solution or more than one. In this case $\vec x \: \in [-1,1]^n$.