Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. For example, $f(x,y) = a_1x^2y^3 + a_2xy^2 + a_3$ and $g(x,y) = b_1x^2y^3 + b_2xy^2 + b_3$. How may common roots of $f(x,y)=0$ and $g(x,y)=0$ are there? I am not interested in finding the common roots but the number. Can Bézout's Theorem help?
Thanks a lot.