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.
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
[2] 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.
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.
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.
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.
[4] 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.
Since with generic polynomials, there are no points at infinity, and all multiplicities equal one, Bézout's formulation is correct, although his proof does not follow the modern requirements of rigor.
The definitions of multiplicities that was given during the first half of the 20th century involved continuous and infinitesimal deformations.
It follows that the proofs of this period apply only over the field of complex numbers.
[6] Modern studies related to Bézout's theorem obtained different upper bounds to system of polynomials by using other properties of the polynomials, such as the Bernstein–Kushnirenko theorem, or generalized it to a large class of functions, such as Nash functions.
[7] 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, without common divisor of positive degree).
Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E that 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 algebraically closed field, which are defined by n homogeneous polynomials in n + 1 variables, of degrees
The affine case of the theorem is the following statement, that was proven in 1983 by David Masser and Gisbert Wüstholz.
[8] Consider n affine hypersurfaces that are defined over an algebraically closed field by n polynomials in n variables, of degrees
This version is not a direct consequence of the general case, because it is possible to have a finite number of intersection points in the affine space, with infinitely many intersection points at infinity.
For stating the general result, one has to recall that the intersection points form an algebraic set, and that there is a finite number of intersection points if and only if all component of the intersection have a zero dimension (an algebraic set of positive dimension has an infinity of points over an algebraically closed field).
Consider n projective hypersurfaces that are defined over an algebraically closed field by n homogeneous polynomials in
For example: The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality.
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 root of a univariate polynomial, for proving that the multiplicity obtained by deformation equals the multiplicity of the corresponding linear factor of the polynomial, one has to know that the roots are continuous functions of the coefficients.
In the case of Bézout's theorem, the general intersection theory can be avoided, as there are proofs (see below) that associate to each input data for the theorem a polynomial in the coefficients of the equations, which factorizes into linear factors, each corresponding to a single intersection point.
The proof that this multiplicity equals the one that is obtained by deformation, results then from the fact that the intersection points and the factored polynomial depend continuously on the roots.
By collecting together the powers of one indeterminate, say y, one gets univariate polynomials whose coefficients are homogeneous polynomials in x and t. For technical reasons, one must change of coordinates in order that the degrees in y of P and Q equal their total degrees (p and q), and each line passing through two intersection points does not pass through the point (0, 1, 0) (this means that no two point have the same Cartesian x-coordinate.
If one defines the multiplicity of a common zero of P and Q as the number of occurrences of the corresponding factor in the product, Bézout's theorem is thus proved.
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 continuous functions of the coefficients of P and Q.
Although a multivariate polynomial is generally irreducible, the U-resultant can be factorized into linear (in the
As for the preceding proof, the equality of this multiplicity with the definition by deformation results from the continuity of the U-resultant as a function of the coefficients of the