Let us assume the following $2\times 2$ system of nonlinear equations $$\begin{split} f_1(x,y)\cdot x+f_2(x,y)\cdot y=f_3(x,y)\\ g_1(x,y)\cdot x+g_2(x,y)\cdot y=g_3(x,y)\ \end{split}$$ for the real variables $x,y$. The functions $f_i,g_i$ are general real polynomials of the form $$f_i,g_i\sim\sum_{n=0}^Nc_nx^{n}y^{N-n}$$ for some $N\in\mathbb{N}$. I was wondering the following:
- Is there a general method of determining whether or not there exist real solutions for this problem? I read somewhere about Hilbert's Nullstellensatz method but I am not able to understand how it applies in this simple setting (or if it is at all relevant).
- If such a method exists and applies here, does it still work when we choose the polynomials to be of infinite degree? Could I assume that they are special functions, e.g. square roots or exponentials, or they have to be polynomials of finite degree?