Bézout's theorem

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Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees. This statement must be qualified in several important ways, by considering points at infinity, allowing complex coordinates (or more generally, coordinates from the algebraic closure of the ground field), assigning an appropriate multiplicity to each intersection point, and excluding a degenerate case when X and Y have a common component. A simpler special case is that if X and Y are both real or complex irreducible curves, X has degree m and Y has degree n then the number of intersection points does not exceed mn.

More generally, number of points in the intersection of 3 algebraic surfaces in projective space is, counting multiplicities, the product of the degrees of the equations of the surfaces, and so on.

Contents

Rigorous statement

Suppose that X and Y are 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.

History

Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his principia, where he claims that two curves have a number of intersection points given by the product of their degrees. The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given.[1]

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