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.
Best Answer
I reread the previous posts. The answers are very interesting but the question is really hilarious. When we have the same terms in each equation (with different coefficients in a field $K$) then the problem is essentially a linear one. For instance, here, we obtain (generically) a sole solution in $K$ for $x^2y^3,xy^2$. Then a value for $x^2y^4$, then a value for $y$, then a value for $x$.
The moral of the story is that some trivial matters can lead to exciting discussions and take time before downgrading certain issues to stackexchange !