﻿ resultant - English German Dictionary        ## resultant

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 English German
resultant adj. resultierend resultant subst. die Resultante f die Resultierende f resultant subst. das Ergebnis n resultant orbital angular momentum subst.   der Gesamtorbitaldrehimpuls m resultant spin angular momentum subst.   der Gesamtspindrehimpuls m accesses today: 64 289.966 words in the dictionary accesses total: 42.861

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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

## 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 mehr

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