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 $m n$ 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 ${\\rm dim}(A)+{\\rm dim}(B)-k$ , then the degree of $A\\cap B$ is the product of the degrees of $A$ and $B$ .

单词	BezoutsTheoremAlgebraicGeometry
释义	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 $m n$ 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 ${\\rm dim}(A)+{\\rm dim}(B)-k$ , then the degree of $A\\cap B$ is the product of the degrees of $A$ and $B$ .
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