Bézout's Theorem

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Theorem

Let $X$ and $Y$ be two plane projective curves defined over a field $F$ that do not have a common component.

(This condition is true if both $X$ and $Y$ are defined by different irreducible polynomials. In particular, it holds for a pair of "generic" curves.)


Then the total number of intersection points of $X$ and $Y$ with coordinates in an algebraically closed field $E$ which contains $F$, counted with their multiplicities, is equal to the product of the degrees of $X$ and $Y$.


Proof


Source of Name

This entry was named for Étienne Bézout.


Historical Note

Bézout's Theorem was originally published in $1799$ by Étienne Bézout in his Théorie Générale des Équations Algébriques.


Sources