= Solution
<Bézout's theorem> says that two <projective plane> curves over $\mathbb C$ of degrees $d,e$ with no common irreducible component have total <intersection multiplicity> $de$, so at most $de$ distinct intersection points. Let the irreducible curve have reduced homogeneous equation $F$ of positive degree $d$. In characteristic zero some <partial derivative> is nonzero. For $d>1$ this <derivative> has degree $d-1$ and cannot share a component with the irreducible degree-$d$ <polynomial> $F$. Every <singular point> belongs to their intersection. Bézout bounds their number by $d(d-1)$, proving \b[there are finitely many <singular points>]. For $d=1$ a partial is a nonzero constant and there are none.
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