Resultant aus Wikipedia.

Zum Beitrag
Resultant - Wikipedia, the free encyclopedia
a.new,#quickbar a.new{color:#ba0000}
/* cache key: enwiki:resourceloader:filter:minify-css:5:f2a9127573a22335c2a9102b208c73e7 */

Resultant
From Wikipedia, the free encyclopedia
Jump to: ,
This article is about the resultant of polynomials. For the result of adding two or more

vectors, see

Parallelogram rule. For the technique in organ building, see

Resultant (organ).
In

mathematics, the resultant of two

monic polynomials P and Q over a

field k is defined as the

product
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

## Contents

## Computation

The resultant is the

determinant of the

Sylvester matrix (and of the

Bezout matrix).
When Q is

separable, the above product can be rewritten to
and this expression remains unchanged if Q is reduced modulo P. Note that, when non-monic, this includes the factor qdegP but still needs the factor pdegQ.
Let . 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 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.
Continuing this procedure ends up in a variant of the

Euclidean algorithm. This procedure needs quadratic runtime.

## Properties

If P' = P + R * Q and degP' = degP, then res(P,Q) = res(P',Q)
If X,Y,P,Q have the same degree and ,
then
res(P ? ,Q) = res(Q ? ,P) where P ? (z) = P( ? z)

## Applications

If x and y are

algebraic numbers such that P(x) = Q(y) = 0 (with degree of Q=n), we see that z = x + y is a root of the resultant (in x) of P(x) and Q(z ? x) and that t = xy is a root of the resultant of P(x) and xnQ(t / x) ; combined with the fact that 1 / y is a root of ynQ(1 / y), this shows that the set of algebraic numbers is a field.
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 . 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

computer algebra, the resultant is a tool that can be used to analyze modular images of the

mehrResultant aus Wikipedia.

Zum Beitrag