Resultant

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Resultant can also refer to the result of adding two or more vectors.

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

<math>\mathrm{res}(P,Q) = \prod_{P(x)=0} \prod_{Q(y)=0} (y-x),\,</math>

of the differences of their roots, where <math>x</math> and <math>y</math> take on values in the algebraic closure of <math>k</math>. For non-monic polynomials with leading coefficients <math>p</math> and <math>q</math>, respectively, the above product is multiplied by

1 Computation
2 Properties
3 Applications
4 References

[edit] Computation

The resultant is the determinant of the Sylvester matrix.

The above product can be rewritten to

<math>\mathrm{res}(P,Q) = \prod_{Q(y)=0} P(y)\,</math>

and this expression remains unchanged if <math>P</math> is reduced modulo <math>Q</math>.

Let <math>P' = P \mod Q</math>. The above idea can be continued by swapping the roles of <math>P'</math> and <math>Q</math>. However, <math>P'</math> has a set of roots different from that of <math>P</math>. This can be resolved by writing <math>\prod_{Q(y)=0} P'(y)\,</math> as a determinant again, where <math>P'</math> has leading zero coefficients. This determinant can now be simplified by iterative expansion with respect to the column, where only the leading coefficient <math>q</math> of <math>Q</math> appears.

<math>\mathrm{res}(P,Q) = q^{\deg P - \deg P'} \cdot \mathrm{res}(P',Q)</math>

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

[edit] Properties

<math>\mathrm{res}(P,Q) = (-1)^{\deg P \cdot \deg Q} \cdot \mathrm{res}(Q,P)</math>
<math>\mathrm{res}(P\cdot R,Q) = \mathrm{res}(P,Q) \cdot \mathrm{res}(R,Q)</math>
If <math>P' = P + R*Q</math> and <math>\deg P' = \deg P</math>, then <math>\mathrm{res}(P,Q) = \mathrm{res}(P',Q)</math>
If <math>X, Y, P, Q</math> have the same degree and <math>X = a_{00}\cdot P + a_{01}\cdot Q, Y = a_{10}\cdot P + a_{11}\cdot Q</math>,

then <math>\mathrm{res}(X,Y) = \det{\begin{pmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{pmatrix}}^{\deg P} \cdot \mathrm{res}(P,Q)</math>

<math>\mathrm{res}(P_-,Q) = \mathrm{res}(Q_-,P)</math> where <math>P_-(z) = P(-z)</math>

[edit] Applications

Resultants can be used in algebraic geometry to determine intersections. For example, let

and

define algebraic curves in <math>\mathbb{A}^2_k</math>. If <math>f</math> and <math>g</math> are viewed as polynomials in <math>x</math> with coefficients in <math>k(y)</math>, then the resultant of <math>f</math> and <math>g</math> gives a polynomial in <math>y</math> whose roots are the <math>y</math>-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 <math>p</math>. 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.

In pipe organs, a resultant is used to offset the extremely expensive cost of a large set of pipes (usually 64'). Whereas a set of 64' pipes could be both cost prohibitive and space prohibitive, a resultant could be more easily be applied. The 64' sound is emulated by having two 32' pipes slightly tuned differently, using the resultant formula above.