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 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. 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.
This and the fact that the concept of intersection multiplicity was outside the knowledge of his time led to a sentiment expressed by some authors that his proof was neither correct nor the first proof to be given.
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, which are not multiples of a common non constant polynomial; 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.
There are various proofs of this theorem, which either are expressed in purely algebraic terms, or use the language or algebraic geometry. Three algebraic proofs are sketched below.
Bézout's theorem has been generalized as the so-called multi-homogeneous Bézout theorem.
The equation of a line in a Euclidean plane is linear, that is, it equates to zero a polynomial of degree one. So, the Bézout bound for two lines is 1, meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are parallel and meet at a point at infinity.
If t is viewed as the coordinate of infinity, a factor equal to t represents an intersection point at infinity.
If at least one partial derivative of the polynomial p is not zero at an intersection point, then the tangent of the curve at this point is defined (see Algebraic curve § Tangent at a point), and the intersection multiplicity is greater than one if and only if the line is tangent to the curve. If all partial derivatives are zero, the intersection point is a singular point, and the intersection multiplicity is at least two.
Two conic sections generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:
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. 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. This number is two in general (ordinary points), but may be higher (three for inflection points, four for undulation points, etc.). This number is the "multiplicity of contact" of the tangent.
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. Deformations cannot be used over fields of positive characteristic. Moreover, there are cases where a convenient deformation is difficult to define (as in the case of more than two planes curves have a common intersection point), and even cases where no deformation is possible.
Presently, following Jean-Pierre Serre, a multiplicity is generally defined as the length of a local ring associated with the point where the multiplicity is considered. Most specific definitions can be shown to be special case of Serre's definition.
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. So, the multiplicity of an intersection point is the multiplicity of the corresponding factor. 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.
Let P and Q be two homogeneous polynomials in the indeterminates x, y, t of respective degrees p and q. Their zeros are the homogeneous coordinates of two projective curves. Thus the homogeneous coordinates of their intersection points are the common zeros of P and Q.
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.
As R is a homogeneous polynomial in two indeterminates, the fundamental theorem of algebra implies that R is a product of pq linear polynomials. 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.
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.
In the early 20th century, Francis Sowerby Macaulay introduced the multivariate resultant (also known as Macaulay's resultant) of n homogeneous polynomials in n indeterminates, which is generalization of the usual resultant of two polynomials. Macaulay's resultant is a polynomial function of the coefficients of n homogeneous polynomials that is zero if and only the polynomials have a nontrivial (that is some component is nonzero) common zero in an algebraically closed field containing the coefficients.
This proof of Bézout's theorem seems the oldest proof that satisfies the modern criteria of rigor.
Bézout's theorem can be proved by recurrence on the number of polynomials by using the following theorem.
Beside allowing a conceptually simple proof of Bézout's theorem, this theorem is fundamental for intersection theory, since this theory is essentially devoted to the study of intersection multiplicities when the hypotheses of the above theorem do not apply.