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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 when one does not care about multiplicities and ''X'' and ''Y'' are two algebraic curves in the Euclidean plane whose implicit equations are polynomials of degrees ''m'' and ''n'' without any non-constant common factor; then the number of intersection points does not exceed ''mn''. Bézout's theorem refers also to the generalization to higher dimensions: Let there be ''n'' homogeneous polynomials in variables, of degrees , that define ''n'' hypersurfaces in the projective space of dimension ''n''. If the number of intersection points of the hypersurfaces is finite over an algebraic closure of the ground field, then this number is if the points are counted with their multiplicity. As in the case of two variables, in the case of affine hypersurfaces, and when not counting multiplicities nor non-real points, this theorem provides only an upper bound of the number of points, which is often reached. This is often referred to as Bézout's bound. Bézout's theorem is fundamental in computer algebra and effective algebraic geometry, by showing that most problems have a computational complexity that is at least exponential in the number of variables. It follows that in these areas, the best complexity that may be hoped for will occur in algorithms have a complexity which is polynomial in Bézout's bound. ==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 means that ''X'' and ''Y'' are defined by polynomials, whose polynomial greatest common divisor is a constant; 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''. The generalization in higher dimension may be stated as: Let ''n'' projective hypersurfaces be given in a projective space of dimension ''n'' over an algebraic closed field, which are defined by ''n'' homogeneous polynomials in ''n'' + 1 variables, of degrees Then either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product If the hypersurfaces are irreducible and in relative general position, then there are intersection points, all with multiplicity 1. There are various proofs of this theorem. In particular, it may be deduced by applying iteratively the following generalization: if ''V'' is a projective algebraic set of dimension and degree , and ''H'' is a hypersurface (defined by a polynomial) of degree , that does not contain any irreducible component of ''V'', then the intersection of ''V'' and ''H'' has dimension and degree For a (sketched) proof using the Hilbert series see Hilbert series and Hilbert polynomial#Degree of a projective variety and Bézout's theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bézout's theorem」の詳細全文を読む スポンサード リンク
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