Resultant
Encyclopedia
In mathematics
, the resultant of two monic polynomials
and 
over a field
is defined as the product
of the differences of their roots, where
and 
take on values in an algebraic closure
of
. For non-monic polynomials with leading coefficients 
and 
, respectively, the above product is multiplied by
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the resultant of two monic polynomials
and over a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
is defined as the productProduct (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
of the differences of their roots, where
and take on values in an algebraic closureAlgebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of
. For non-monic polynomials with leading coefficients and , respectively, the above product is multiplied byComputation
- The resultant is the determinantDeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the Sylvester matrixSylvester matrixIn mathematics, a Sylvester matrix is a matrix associated to two polynomials that provides information about those polynomials. It is named for James Joseph Sylvester.-Definition:...
(and of the Bézout matrixBézout matrixIn mathematics, a Bézout matrix is a special square matrix associated with two polynomials. Such matrices are sometimes used to test the stability of a given polynomial.-Definition:...
).
- When Q is separableSeparable polynomialIn mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
, the above product can be rewritten to
- and this expression remains unchanged if
is reduced modulo 
. Note that, when non-monic, this includes the factor 
but still needs the factor 
.
- Let
. The above idea can be continued by swapping the roles of 
and 
. However, 
has a set of roots different from that of 
. This can be resolved by writing 
as a determinant again, where 
has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient 
of 
appears.
- Continuing this procedure ends up in a variant of the Euclidean algorithmEuclidean algorithmIn mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
. This procedure needs quadratic runtime.
Properties
- The resultant is a polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
with integer coefficients in term of the coefficients of
and 
. It follows that
- The resultant is well defined for polynomials over any commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. - If h is a homomorphismHomomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
of the ring of the coefficients into another commutative ring, which preserve the degrees of
and 
, then the resultant of the image by h of 
and 
is the image by h of the resultant of 
and 
.
- The resultant is well defined for polynomials over any commutative ring
- The resultant of two polynomials with coefficient in a field is null if and only if they have a GCDGreatest common divisor of two polynomialsInformally, the greatest common divisor of two polynomials p and q is the largest polynomial that divides both p and q evenly. The definition is modeled on the concept of the greatest common divisor of two integers, the greatest integer that divides both...
of positive degree. -
-
- If
and 
, then 
- If
have the same degree and 
,
- then
-
where 
-
Applications
- If x and y are algebraic numbers such that
(with degree of Q=n), we see that 
is a root of the resultant (in x) of 
and 
and that 
is a root of the resultant of 
and 
; combined with the fact that 
is a root of 
, this shows that the set of algebraic numbers is a field.
- The discriminantDiscriminantIn algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
of a polynomial is the quotient by its leading coefficient of the resultant of the polynomial and its derivative.
- Resultants can be used in algebraic geometryAlgebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
to determine intersections. For example, let
- and
- define algebraic curveAlgebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s in
. If 
and 
are viewed as polynomials in 
with coefficients in 
, then the resultant of 
and 
is a polynomial in 
whose roots are the 
-coordinates of the intersection of the curves and of the common asymptotes parallel to the 
axis.
- In computer algebra, the resultant is a tool that can be used to analyze modular images of the greatest common divisorGreatest common divisorIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
of integer polynomials where the coefficients are taken modulo some prime number
. The resultant of two polynomials is frequently computed in the Lazard-Rioboo-Trager method of finding the integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of a ratio of polynomials.
- In wavelet theory, the resultant is closely related to the determinant of the transfer matrixTransfer matrixThe transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory....
of a refinable function.