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 the
algebraic closureof . For non-monic polynomials with leading coefficients and , respectively, the above product is multiplied by
* The resultant is the
determinantof 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 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 algorithm. This procedure needs quadratic runtime.
* If and , then
* If have the same degree and ,:then
* The resultant of a polynomial and its derivative is related to the
* Resultants can be used in
algebraic geometryto determine intersections. For example, let : :and : :define algebraic curves in . If and are viewed as polynomials in with coefficients in , then the resultant of and gives a polynomial in whose roots are the -coordinates of the intersection of the curves.
Galois theory, resultants can be used to compute norms.
computer algebra, the resultant is a tool that can be used to analyze modular images of the greatest common divisorof integer polynomials where the coefficients are taken modulo some prime number . The resultant of two polynomials is frequently computed in the Lazard-Rioboo-Trager methodof finding the integralof a ratio of polynomials.
* In wavelet theory, the resultant is closely related to the determinant of the
transfer matrixof a refinable function.
* [http://mathworld.wolfram.com/Resultant.html Weisstein, Eric W. "Resultant." From MathWorld--A Wolfram Web Resource.]
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