A line meets a degree-d curve in d points counted with multiplicity

Given: AC The Axiom of Choice, a line $L=V(\mathrm{\ell })$ and a plane projective curve $C=V(F)$ of degree $d$ over the algebraically closed field $k$, with $L\subseteq ̸C$.

$L$ is a plane projective curve of degree one, and $L,C$ have no common component; so Bezout applies to the pair and gives ${\sum }_{p\in L\cap C}{I}_p(C,L)=d\cdot 1=d$, the sum being finite Plane projective curves and their components, degree projective hypersurface, Bezout's theorem for plane projective curves.

The restriction $F{\mathrm{\mid }}_L$ is a nonzero binary form of degree $d$ on $L\cong {\mathbf{P}}^1$ (nonzero because $L\subseteq ̸C$), and $I_p(C,L)$ equals the order of vanishing of $F{\mathrm{\mid }}_L$ at the point corresponding to $p$ Intersection with a line is the order of vanishing of the restricted equation, projective space points, standard projective opens are affine spaces.

A nonzero binary form of degree $d$ over an algebraically closed field is a product of $d$ linear forms, so the multiplicities of its distinct roots sum to $d$ by additivity of degree over products Tangent cone and tangent lines at a point (Remarks, binary-form factorisation); the homogeneous-product degree calculation there gives the count.