Degree theory and systems of nonlinear algebraic equations

Question

For a system of nonlinear algebraic equations, how to find the number of solutions to this system? Any related degree theory can be used to determine the number of solutions? Are there any recommended references?

Answer 1

The key result about this problem is the Bézout's theorem, a fundamental result in algebraic geometry.

The simpler formulation of this theorem states that:

if $P(x,y)$ and $Q(x,y)$ are two polynomials of degree $m$ and $n$ over a field $\mathbb{F}$, whose GCD polynomial is a constant, then the points in the set $Z(P,Q)=\{(x,y) \in \mathbb{F}^2 :P(x,y)=0 \land Q(x,y)=0 \}$ are at most $mn$.

The statement can be made stronger if we consider the multiplicities of solutions an the solutions at ''infinity'' in the projective plane. In this case the number of solutions is exactly $nm$. But the definition of the multiplicity require some attention.

We can also extend the theorem to higher dimensions, i.e. to polynomial in more than two variables, but the definition of multiplicity becomes more complex.

For an introduction you can see here or here.

Answer 2

Thank you very much for pointing out the two elegant references (especially Bezout’s Theorem- A taste of algebraic geometry.)

I just took a glance at them. I've not studied algebraic geometry, but it seems algebraic geometry is interesting and can solve my problem such as:

\begin{equation} \begin{cases} \sin u+4\,u+7\,v+2\,u^2+5\,v^2=3, \\ \cos v+9\,u+5\,v+8\,u^2+2\,v^2=6. \\ \end{cases} \end{equation} Could we determine the number of solutions $(u,v)$ (if any) to the above nonlinear algebraic equations (not in the form of polynomials) applying Bezout’s Theorem?

I'm not sure if my understanding of Bezout’s Theorem is correct or not.