Resultant

This article is about the resultant of polynomials. For the result of adding two or more vectors, see Parallelogram rule. For the musical phenomenon, see Resultant tone.

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

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

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 $P$ and $Q$, 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 $P' = P + R*Q$ and $\deg P' = \deg P$, then $\mathrm{res}(P,Q) = \mathrm{res}(P',Q)$
• If $X, Y, P, Q$ have the same degree and $X = a_{00}\cdot P + a_{01}\cdot Q, Y = a_{10}\cdot P + a_{11}\cdot Q,$

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

• $\mathrm{res}(P_-,Q) = \mathrm{res}(Q_-,P)\,$ 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 $z=x+y$ 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.

See also

• Elimination theory

References

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