When does a system of two bivariate polynomials not have roots at infinity? I have no background in algebraic geometry, and I have the following (probably) basic question. Let $p,q\in\mathbb{C}[x,y]$ be polynomials of degree $d$, and consider the  system:
$$ p(x,y) = 0, \qquad q(x,y) = 0. $$
This system has a common root at infinity if the corresponding homogenized system:
$$ p_h(t,x,y) = t^d p(x/t,y/t)=0, \qquad q_h(t,x,y) = t^d q(x/t,y/t)=0 $$
has a solution $(t^*,x^*,y^*)$ with $t^*=0$. Does there exists a simple criterion on the polynomial coefficients which states when something like this cannot occur?

Yes. If the homogeneous polynomials which are the highest-degree portions of $p$ and $q$ are coprime, there is no common root at infinity. One way you can test this coprimality without finding any roots of these polynomials is by computing the homogeneous resultant of the top-degree terms: if the resultant is nonzero, the top-degree terms are coprime and have no common root at infinity.
