Definitions

Resultant

[ri-zuhl-tnt]

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

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

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

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

Computation

$mathrm\left\{res\right\}\left(P,Q\right) = prod_\left\{P\left(x\right)=0\right\} Q\left(x\right),$
and this expression remains unchanged if $Q$ is reduced modulo $P$. Note that, when non-monic, this includes the factor $q^\left\{deg P\right\}$ but still needs the factor $p^\left\{deg Q\right\}$.

• Let $P\text{'} = P mod Q$. The above idea can be continued by swapping the roles of $P\text{'}$ and $Q$. However, $P\text{'}$ has a set of roots different from that of $P$. This can be resolved by writing $prod_\left\{Q\left(y\right)=0\right\} P\text{'}\left(y\right),$ as a determinant again, where $P\text{'}$ 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.

$mathrm\left\{res\right\}\left(P,Q\right) = q^\left\{deg P - deg P\text{'}\right\} cdot mathrm\left\{res\right\}\left(P\text{'},Q\right)$
Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

Properties

• $mathrm\left\{res\right\}\left(P,Q\right) = \left(-1\right)^\left\{deg P cdot deg Q\right\} cdot mathrm\left\{res\right\}\left(Q,P\right)$
• $mathrm\left\{res\right\}\left(Pcdot R,Q\right) = mathrm\left\{res\right\}\left(P,Q\right) cdot mathrm\left\{res\right\}\left(R,Q\right)$
• If $P\text{'} = P + R*Q$ and $deg P\text{'} = deg P$, then $mathrm\left\{res\right\}\left(P,Q\right) = mathrm\left\{res\right\}\left(P\text{'},Q\right)$
• If $X, Y, P, Q$ have the same degree and $X = a_\left\{00\right\}cdot P + a_\left\{01\right\}cdot Q, Y = a_\left\{10\right\}cdot P + a_\left\{11\right\}cdot Q$,

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

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

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

$f\left(x,y\right)=0$
and
$g\left(x,y\right)=0$
define algebraic curves in $mathbb\left\{A\right\}^2_k$. If $f$ and $g$ are viewed as polynomials in $x$ with coefficients in $k\left(y\right)$, then the resultant of $f$ and $g$ gives a polynomial in $y$ whose roots are the $y$-coordinates of the intersection of the curves.