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Concept
Bézout's theorem
Summary
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that in general the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout.
In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two plane algebraic curves of degrees and have no component in common, they have intersection points, counted with their multiplicity, and including points at infinity and points with complex coordinates.
In its modern formulation, the theorem states that, if N is the number of common points over an algebraically closed field of n projective hypersurfaces defined by homogeneous polynomials in n + 1 indeterminates, then N is either infinite, or equals the product of the degrees of the polynomials. Moreover, the finite case occurs almost always.
In the case of two variables and in the case of affine hypersurfaces, if multiplicities and points at infinity are not counted, this theorem provides only an upper bound of the number of points, which is almost always reached. This bound is often referred to as the Bézout 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 can be hoped for will occur with algorithms that have a complexity which is polynomial in the Bézout bound.
In the case of plane curves, Bézout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia in 1687, where he claims that two curves have a number of intersection points given by the product of their degrees.
The general theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. He supposed the equations to be "complete", which in modern terminology would translate to generic.
Official source
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