Bézout's theorem is a statement in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
concerning the number of common
zeros of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s in 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
Étienne Bézout (; 31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France.
Work
In 1758 Bézout was elected an adjoint in mechanics of the ...
.
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
coordinates.
In its modern formulation, the theorem states that, if is the number of common points over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
of
projective hypersurfaces defined by
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s in indeterminates, then 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
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
and effective
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, by showing that most problems have a
computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...
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.
History
In the case of plane curves, Bézout's theorem was essentially stated by
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
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
Étienne Bézout (; 31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France.
Work
In 1758 Bézout was elected an adjoint in mechanics of the ...
's ''Théorie générale des équations algébriques''. He supposed the equations to be "complete", which in modern terminology would translate to
generic
Generic or generics may refer to:
In business
* Generic term, a common name used for a range or class of similar things not protected by trademark
* Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
. 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.
The proof of the statement that includes multiplicities was not possible before the 20th century with the introduction of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.
Statement
Plane curves
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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
''E'' which contains ''F'', counted with their
multiplicities, is equal to the product of the degrees of ''X'' and ''Y''.
General case
The generalization in higher dimension may be stated as:
Let ''n''
projective hypersurfaces be given in a
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
of dimension ''n'' over an algebraically closed field, which are defined by ''n''
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s in ''n'' + 1 variables, of degrees
Then either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product
If the hypersurfaces are irreducible and in relative
general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
, then there are
intersection points, all with multiplicity 1.
There are various proofs of this theorem, which either are expressed in purely algebraic terms, or use the language or
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. Three algebraic proofs are sketched below.
Bézout's theorem has been generalized as the so-called
multi-homogeneous Bézout theorem.
Examples (plane curves)
Two lines
The equation of a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
in a
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
is
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
, that is, it equates to zero a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
of degree one. So, the Bézout bound for two lines is , meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster o ...
and meet at a
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
.
One can verify this with equations. The equation of a first line can be written in
slope-intercept form
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
or, in
projective coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinate system, Cartesian coordinates are u ...
(if the line is vertical, one may exchange and ). If the equation of a second line is (in projective coordinates)
by substituting
for in it, one gets
If
one gets the -coordinate of the intersection point by solving the latter equation in and putting
If
that is
the two line are parallel as having the same slope. If
they are distinct, and the substituted equation gives . This gives the point at infinity of projective coordinates .
A line and a curve
As above, one may write the equation of the line in projective coordinates as
If curve is defined in projective coordinates by a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
of degree , the substitution of provides a homogeneous polynomial of degree in and . The
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
implies that it can be factored in linear factors. Each factor gives the ratio of the and coordinates of an intersection point, and the multiplicity of the factor is the multiplicity of the intersection point.
If is viewed as the ''coordinate of infinity'', a factor equal to represents an intersection point at infinity.
If at least one partial derivative of the polynomial is not zero at an intersection point, then the tangent of the curve at this point is defined (see ), 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
Singularity or singular point may refer to:
Science, technology, and mathematics Mathematics
* Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
, and the intersection multiplicity is at least two.
Two conic sections
Two
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
s 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:
*Two circles never intersect in more than two points in the plane, while Bézout's theorem predicts four. The discrepancy comes from the fact that every circle passes through the same two complex points on the line at infinity. Writing the circle
in
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
, we get
from which it is clear that the two points and lie on every circle. When two circles do not meet at all in the real plane, the two other intersections have non-zero imaginary parts, or if they are concentric then they meet at exactly the two points on the line at infinity with an intersection multiplicity of two.
*Any conic should meet the line at infinity at two points according to the theorem. A hyperbola meets it at two real points corresponding to the two directions of the asymptotes. An ellipse meets it at two complex points which are conjugate to one another---in the case of a circle, the points and . A parabola meets it at only one point, but it is a point of tangency and therefore counts twice.
*The following pictures show examples in which the circle meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than one:
Multiplicity
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 point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s, four for
undulation point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
s, 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
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of a
univariate polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
, 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 function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s 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.
Currently, following
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, a multiplicity is generally defined as the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
of a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
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
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
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.
Proofs
Using the resultant (plane curves)
Let and be two homogeneous polynomials in the indeterminates of respective degrees and . Their zeros are the
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
of two
projective curve
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
s. Thus the homogeneous coordinates of their intersection points are the common zeros of and .
By collecting together the powers of one indeterminate, say , one gets univariate polynomials whose coefficients are homogeneous polynomials in and .
For technical reasons, one must
change of coordinates
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consi ...
in order that the degrees in of and equal their total degrees ( and ), and each line passing through two intersection points does not pass through the point (this means that no two point have the same
Cartesian -coordinate.
The
resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ov ...
of and with respect to is a homogeneous polynomial in and that has the following property:
with
if and only if it exist
such that
is a common zero of and (see ). The above technical condition ensures that
is unique. The first above technical condition means that the degrees used in the definition of the resultant are and ; this implies that the degree of is (see ).
As is a homogeneous polynomial in two indeterminates, the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
implies that is a product of linear polynomials. If one defines the multiplicity of a common zero of and 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 function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s of the coefficients of and .
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.
Using -resultant
In the early 20th century,
Francis Sowerby Macaulay introduced the
multivariate resultant (also known as ''Macaulay's resultant'') of
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s in indeterminates, which is generalization of the usual
resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ov ...
of two polynomials. Macaulay's resultant is a polynomial function of the coefficients of 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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
containing the coefficients.
The -resultant is a particular instance of Macaulay's resultant, introduced also by Macaulay. Given homogeneous polynomials
in indeterminates
the -resultant is the resultant of
and
where the coefficients
are auxiliary indeterminates. The -resultant is a homogeneous polynomial in
whose degree is the product of the degrees of the
Although a multivariate polynomial is generally
irreducible, the -resultant can be factorized into linear (in the
) polynomials over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
containing the coefficients of the
These linear factors correspond to the common zeros of the
in the following way: to each common zero
corresponds a linear factor
and conversely.
This proves Bézout's theorem, if the multiplicity of a common zero is defined as the multiplicity of the corresponding linear factor of the -resultant. As for the preceding proof, the equality of this multiplicity with the definition by deformation results from the continuity of the -resultant as a function of the coefficients of the
This proof of Bézout's theorem seems the oldest proof that satisfies the modern criteria of rigor.
Using the degree of an ideal
Bézout's theorem can be proved by recurrence on the number of polynomials
by using the following theorem.
''Let be a
projective algebraic set
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
and
degree , and be a hypersurface (defined by a single polynomial) of degree
, that does not contain any
irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal ( ...
of ; under these hypotheses, the intersection of and has dimension
and degree
''
For a (sketched) proof using
Hilbert series
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homog ...
, see .
Beside allowing a conceptually simple proof of Bézout's theorem, this theorem is fundamental for
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
, since this theory is essentially devoted to the study of intersection multiplicities when the hypotheses of the above theorem do not apply.
See also
*
*
Notes
References
*
* Alternative translation of earlier (2nd) edition of Newton's ''Principia''.
* (generalization of theorem) https://mathoverflow.net/q/42127
External links
*
*
Bezout's Theorem at MathPages
{{DEFAULTSORT:Bezouts Theorem
Theorems in plane geometry
Incidence geometry
Intersection theory
Theorems in algebraic geometry