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 an algebraic closure of $k$ , and are repeated according to their multiplicities as roots of the polynomials. 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

Computation

For a fixed polynomial $P$ , the above product can be rewritten as

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

so it can be expressed (polynomially) in terms of the coefficients of $Q$ . Another way to see this is to notice that $res(P,Q)$ depends polynomially (with integer coefficients) on the roots of $P$ and $Q$ , and it is invariant with respect to permutations of each set of roots, so it must be possible to calculate it using an (integer) polynomial formula on the coefficients of $P$ and $Q$ . See elementary symmetric polynomial for details.

More concretely, the resultant is the determinant of the Sylvester matrix (and of the Bézout matrix) associated to $P$ and $Q$ .

The expression

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

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

Since the resultant is a polynomial with integer coefficients in term of the coefficients of and , it follows that
- The resultant is well defined for polynomials over any commutative ring.
- If h is a homomorphism of the ring of the coefficients into another commutative ring, which preserve the degrees of $P$ and $Q$ , then the resultant of the image by h of $P$ and $Q$ is the image by h of the resultant of $P$ and $Q$ .
The resultant of two polynomials with coefficients in a integral domain is null if and only if they have a common divisor of positive degree.
$\mathrm{res}(P,Q) = (-1)^{\deg P \cdot \deg Q} \cdot \mathrm{res}(Q,P)$
$\mathrm{res}(P\cdot R,Q) = \mathrm{res}(P,Q) \cdot \mathrm{res}(R,Q)$
If $P' = P %2B R*Q$ and $\deg P' = \deg P$ , then $\mathrm{res}(P,Q) = \mathrm{res}(P',Q)$
If $X, Y, P, Q$ have the same degree and $X = a_{00}\cdot P %2B a_{01}\cdot Q, Y = a_{10}\cdot P %2B a_{11}\cdot Q$ ,

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

$\mathrm{res}(P_-,Q) = \mathrm{res}(Q_-,P)$ where $P_-(z) = P(-z)$

Applications

If x and y are algebraic numbers such that $P(x)=Q(y)=0$ (with degree of Q=n), we see that $z=x%2By$ is a root of the resultant (in x) of $P(x)$ and $Q(z-x)$ and that $t=xy$ is a root of the resultant of $P(x)$ and $x^nQ(t/x)$ ; combined with the fact that $1/y$ is a root of $y^nQ(1/y)$ , this shows that the set of algebraic numbers is a field.

The discriminant of a polynomial is the quotient by its leading coefficient of the resultant of the polynomial and its derivative.

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 $g$ is a polynomial in $y$ whose roots are the $y$ -coordinates of the intersection of the curves and of the common asymptotes parallel to the $x$ axis.

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

Weisstein, Eric W., "Resultant" from MathWorld.

Resultant

Contents

Computation

Properties

Applications

See also

References