Resultant

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This article is about the resultant of polynomials. For the result of adding two or more vectors, see Parallelogram rule. For the technique in organ building, see Resultant (organ).

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

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

1 Computation
2 Properties
3 Applications
4 See also
5 References

[edit] 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.

[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

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.