Resultant

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

[\mathrm(P,Q) = \prod_ \prod_ (y-x),\,]

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^ q^.\,]

Contents

1 Computation
2 Properties
3 Applications
4 References

Computation

The resultant is the determinant of the Sylvester matrix.

The above product can be rewritten to

[\mathrm(P,Q) = \prod_ P(y)\,]

and this expression remains unchanged if [P] is reduced modulo [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_ 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(P,Q) = q^ \cdot \mathrm(P',Q)]

Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

Properties

[\mathrm(P,Q) = (-1)^ \cdot \mathrm(Q,P)]
[\mathrm(P\cdot R,Q) = \mathrm(P,Q) \cdot \mathrm(R,Q)]
If [P' = P + R*Q] and [\deg P' = \deg P], then [\mathrm(P,Q) = \mathrm(P',Q)]
If [X, Y, P, Q] have the same degree and [X = a_\cdot P + a_\cdot Q, Y = a_\cdot P + a_\cdot Q],

then [\mathrm(X,Y) = \det a_ & a_ \\ a_ & a_ \end}^ \cdot \mathrm(P,Q)]

[\mathrm(P_-,Q) = \mathrm(Q_-,P)] where [P_-(z) = P(-z)]

Applications

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^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.

References

[Mathworld entry]

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