NationMaster - Encyclopedia: Resultant

In mathematics, the resultant of two monic polynomials $P$ and $Q$ over a field $k$ is defined as the product In pipe organs, a resultant is a combination of pipes which allow the listener to hear a lower pitched sound. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

$mathrm{res}(P,Q) = prod_{P(x)=0} prod_{Q(y)=0} (y-x),,$

of the differences of their roots, where $x$ and $y$ take on values in the algebraic closure of $k$ . For non-monic polynomials with leading coefficients $p$ and $q$ , respectively, the above product is multiplied by In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...

$p^{deg Q} q^{deg P}.,$

1 Computation
2 Properties
3 Applications
4 References

Computation

The resultant is the determinant of the Sylvester matrix.

The above product can be rewritten to

$mathrm{res}(P,Q) = prod_{Q(y)=0} P(y),$

and this expression remains unchanged if

P

is reduced modulo

Q

Let $P' = P mod Q$ . The above idea can be continued by swapping the roles of $P'$ and $Q$ . However, $P'$ has a set of roots different from that of $P$ . This can be resolved by writing $prod_{Q(y)=0} P'(y),$ as a determinant again, where $P'$ has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient $q$ of $Q$ appears.

$mathrm{res}(P,Q) = q^{deg P - deg P'} cdot mathrm{res}(P',Q)$

Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a Sylvester matrix is a matrix associated to two polynomials that gives us some information about those polynomials. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...

Properties

$mathrm{res}(P,Q) = (-1)^{deg P cdot deg Q} cdot mathrm{res}(Q,P)$
$mathrm{res}(Pcdot R,Q) = mathrm{res}(P,Q) cdot mathrm{res}(R,Q)$
If $P' = P + R * Q$ and $deg P' = deg P$ , then $res(P, Q) = res(P', Q)$
If $X, Y, P, Q$ have the same degree and $X = a_{00}cdot P + a_{01}cdot Q, Y = a_{10}cdot P + a_{11}cdot Q$ ,

then $mathrm{res}(X,Y) = det{begin{pmatrix} a_{00} & a_{01} a_{10} & a_{11} end{pmatrix}}^{deg P} cdot mathrm{res}(P,Q)$

$res(P -, Q) = res(Q -, P)$ where $P - (z) = P ( - z)$

Applications

The resultant of a polynomial and its derivative is related to the discriminant.

Resultants can be used in algebraic geometry to determine intersections. For example, let

f (x, y) = 0

and

g (x, y) = 0

define algebraic curves in $mathbb{A}^2_k$ . If

f

and

g

are viewed as polynomials in

x

with coefficients in

k (y)

, then the resultant of

f

and

g

gives a polynomial in

y

whose roots are the

y

-coordinates of the intersection of the curves.

In Galois theory, resultants can be used to compute norms.

In computer algebra, the resultant is a tool that can be used to analyze modular images of the greatest common divisor of integer polynomials where the coefficients are taken modulo some prime number $p$ . The resultant of two polynomials is frequently computed in the Lazard-Rioboo-Trager method of finding the integral of a ratio of polynomials.

In wavelet theory, the resultant is closely related to the determinant of the transfer matrix of a refinable function.

In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ... In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In calculus, the integral of a function is an extension of the concept of a sum. ... All wavelet transforms consider a function (taken to be a function of time) in terms of oscillations which are localised in both time and frequency. ... The transfer matrix is a formulation in terms of a matrix of the two-scale equation, which characterizes refinable functions. ... In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfills some kind of self-similarity. ...

References

Mathworld entry

Categories: Polynomials | Determinants

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