= Resultant bound for intersections of plane curves
{title2=$|C\cap D|\le\deg C\deg D$}
For <projective plane curves> without a common <irreducible component>, project from a point outside both <curves>. In coordinates where that point is $[1:0:0]$, their defining <homogeneous polynomials> have constant nonzero leading coefficients in $X$. The <resultant> in $X$ is nonzero by <Gauss's lemma>, and is homogeneous of degree $\deg C\deg D$ in the remaining two coordinates. Every intersection projects to one of its zeros. Each projection line has only finitely many intersections with the curves, so their intersection is finite. Choose the center again outside all joining lines of pairs of intersection points; the projection is now injective on that finite set, proving the bound.
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