It is named after Étienne Bézout. 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. 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. Bézout a généralisé ce théorème aux polynômes. 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

Le théorème de Bézout a en fait été énoncé par Bachet de Méziriac. v = c admet des solutions entières si et seulement si c est un multiple de d.

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. Théorème de Bezout Soient a et b deux entiers relatifs non nuls. In its original form the theorem states that in general the number of common zeros equals the product of the degrees of the polynomials. Bézout's theorem can be proved by recurrence on the number of polynomials 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.

Les démonstrations ci-dessous fournissent une seule solution, mais il en existe en général une infinité d'autres. 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. 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. 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. For the identity relating two numbers and their greatest common divisor, see Les deux théorèmes assurent l'existence d'un couple d'entiers tels que ax + by = pgcd(a, b). In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two In its modern formulation, the theorem states that, if 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. 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 . by using the following theorem. 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:

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