﻿ resultant - English German Dictionary

## resultant

1-6
 English German
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
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## 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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