Bézout's Theorem
Theorem
Let $X$ and $Y$ be two plane projective curves defined over a field $F$ that do not have a common component.
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
The condition that $X$ and $Y$ have no common component is true if both $X$ and $Y$ are defined by different irreducible polynomials.
In particular, it holds for a pair of "generic" curves.
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Examples
Conic Sections
Two conic sections meet in at most $4$ points.
Also known as
Some sources omit the accent off the name: Bezout's theorem, which may be a mistake.
Source of Name
This entry was named for Étienne Bézout.
Historical Note
Bézout's Theorem was originally published in $1779$ by Étienne Bézout in his Théorie Générale des Équations Algébriques.
Sources
- 1779: Étienne Bézout: Théorie Générale des Équations Algébriques
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Bezout's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bézout's Theorem