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

Parallel Lines

Two parallel straight lines still meet at one point at infinity.


Parabola and Tangent Line

A parabola $\PP$ and a tangent to $\PP$ meet at a point of multiplicity $2$. Two parallel straight lines still meet at one point at infinity.


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

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
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bézout's theorem
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