Number of roots of two polynomials

Question

Let $f,g\in R[X,Y]$ and suppose that $\{ (x,y) \in R^2 : f(x,y)=g(x,y)=0\}$ is finite. If the variable $X$ occurs only up to degree $d_1$ in both $g,f$ and $Y$ only up to degree $d_2$ is there anything that can be said about the number of common roots which is better than $(d_1+d_2)^2$?

Answer 1

This is an application of Bézout's Theorem. If $R$ is a field, $X^{d_1}Y^{d_2}$ appears in both $g$ and $f$ and the curves $Z(g)$ and $Z(f)$ are in sufficiently general position, then you will have $\deg(f)\cdot\deg(g) = (d_1+d_2)^2$ many common roots. So, no, in general there is no better bound.