# Bézout’s theorem (Algebraic Geometry)

The classic version of Bézout’s theorem^{} states that two complex projective curves of degrees $m$ and $n$ which share no common component intersect in exactly $mn$ points if the points are counted with multiplicity^{}.

The generalized version of Bézout’s theorem states that if $A$ and $B$ are algebraic varieties in $k$-dimensional projective space over an algebraically complete field and $A\cap B$ is a variety^{} of dimension $\mathrm{dim}(A)+\mathrm{dim}(B)-k$, then the degree of $A\cap B$ is the product^{} of the degrees of $A$ and $B$.

Title | Bézout’s theorem (Algebraic Geometry^{}) |
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Canonical name | BezoutsTheoremAlgebraicGeometry |

Date of creation | 2013-03-22 14:36:45 |

Last modified on | 2013-03-22 14:36:45 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Algorithm |

Classification | msc 14A10 |