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 $P$ and $Q$ over a field $k$ is defined as the product

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

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

$p^{\deg Q} q^{\deg P}.\,$

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

$\mathrm{res}(P,Q) = \prod_{Q(y)=0} P(y)\,$

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 $\prod_{Q(y)=0} P'(y)\,$ 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.

[edit] Properties

$\mathrm{res}(P,Q) = (-1)^{\deg P \cdot \deg Q} \cdot \mathrm{res}(Q,P)$
$\mathrm{res}(P\cdot R,Q) = \mathrm{res}(P,Q) \cdot \mathrm{res}(R,Q)$
If $P' = P + R * Q$ and $deg P' = deg P$ , then $res(P, Q) = res(P', Q)$
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 $\mathrm{res}(X,Y) = \det{\begin{pmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{pmatrix}}^{\deg P} \cdot \mathrm{res}(P,Q)$

$res(P -, Q) = res(Q -, P)$ where $P - (z) = P ( - z)$

[edit] 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 $\mathbb{A}^2_k$ . 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.

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 playing both the desired note and it's fifth to create the sound of a pitch one octave lower. For example, playing a 32 foot stop CCCC along with GGGG would create the sound of CCCCC.