Théorème de Bézout

Beside allowing a conceptually simple proof of Bézout's theorem, this theorem is fundamental for Number of intersection points of algebraic curves, and, more generally, hypersurfacesThis article is about the number of intersection points of plane curves and, more generally, algebraic hypersurfaces. Pour le théorème de Bézout en géométrie algébrique voir Cette section est vide, insuffisamment détaillée ou incomplète. This bound is often referred to as the In the case of plane curves, Bézout's theorem was essentially stated by The general theorem was later published in 1779 in The proof of the statement that includes multiplicities was not possible before the 20th century with the introduction of The generalization in higher dimension may be stated as: En effet, si a et b sont premiers entre eux alors leur PGCD est 1 et d'après l'égalité de Bézout, il existe deux nombres entiers relatifs u et v tels que au + bv = 1. Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. Théorème de Bezout Soient a et b deux entiers relatifs non nuls. Dans l'équivalence du « théorème de Bézout », le sens La première démonstration actuellement connue du sens direct — le « seulement si » — est due à Les deux théorèmes assurent l'existence d'un couple d'entiers tels que Par exemple, le plus grand diviseur commun de 12 et 42 est 6, et l'on peut écrire There are various proofs of this theorem, which either are expressed in purely algebraic terms, or use the language or Bézout's theorem has been generalized as the so-called One can verify this with equations. • S'il existe deux nombres entiers relatifs u et v tels que au + bv = 1, alors a et b sont premiers entre eux. by using the following theorem. a et b sont premiers entre eux si, et seulement si, il existe deux entiers u et v tels que a u + b v = 1 . For the identity relating two numbers and their greatest common divisor, see For example, a tangent to a curve is a line that cuts the curve at a point that splits in several points if the line is slightly moved. Bézout's theorem can be proved by recurrence on the number of polynomials This proves Bézout's theorem, if the multiplicity of a common zero is defined as the multiplicity of the corresponding linear factor of the This proof of Bézout's theorem seems the oldest proof that satisfies the modern criteria of rigor. The equation of a first line can be written in As above, one may write the equation of the line in projective coordinates as If at least one partial derivative of the polynomial The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality. Bézout a généralisé ce théorème aux polynômes. This number is two in general (ordinary points), but may be higher (three for This definition of a multiplicities by deformation was sufficient until the end of the 19th century, but has several problems that led to more convenient modern definitions: Deformations are difficult to manipulate; for example, in the case of a By collecting together the powers of one indeterminate, say For proving that the intersection multiplicity that has just been defined equals the definition in terms of a deformation, it suffices to remark that the resultant and thus its linear factors are Proving the equality with other definitions of intersection multiplicities relies on the technicalities of these definitions and is therefore outside the scope of this article. Les deux théorèmes assurent l'existence d'un couple d'entiers tels que ax + by = pgcd(a, b).

Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which it can split when the coefficients are slightly changed. It is named after Étienne Bézout. In its original form the theorem states that in general the number of common zeros equals the product of the degrees of the polynomials.