Resultant


Resultant

In mathematics, the resultant of two monic polynomials P and Q over a field k is defined as the product

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

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

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

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:mathrm{res}(P,Q) = prod_{P(x)=0} Q(x),:and this expression remains unchanged if Q is reduced modulo P. Note that, when non-monic, this includes the factor q^{deg P} but still needs the factor p^{deg 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.

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 mathrm{res}(P,Q) = mathrm{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{egin{pmatrix} a_{00} & a_{01} \ a_{10} & a_{11} end{pmatrix^{deg P} cdot mathrm{res}(P,Q)
* mathrm{res}(P_-,Q) = mathrm{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.

See also

Elimination theory

References

* [http://mathworld.wolfram.com/Resultant.html Weisstein, Eric W. "Resultant." From MathWorld--A Wolfram Web Resource.]


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