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 the algebraic closure of $k$. For non-monic polynomials with leading coefficients $p$ and $q$, respectively, the above product is multiplied by

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

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 $P\text{'}\; =\; P\; mod\; Q$. 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 $prod\_\{Q(y)=0\}\; P\text{'}(y),$ 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.:$mathrm\{res\}(P,Q)\; =\; q^\{deg\; P\; -\; deg\; P\text{'}\}\; cdot\; mathrm\{res\}(P\text{'},Q)$: Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

Properties

* $mathrm\{res\}(P,Q)\; =\; (-1)^\{deg\; P\; cdot\; deg\; Q\}\; cdot\; mathrm\{res\}(Q,P)$
* $mathrm\{res\}(Pcdot\; R,Q)\; =\; mathrm\{res\}(P,Q)\; cdot\; mathrm\{res\}(R,Q)$
* If $P\text{'}\; =\; P\; +\; R*Q$ and $deg\; P\text{'}\; =\; deg\; P$, then $mathrm\{res\}(P,Q)\; =\; mathrm\{res\}(P\text{'},Q)$
* 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
* $mathrm\{res\}(P\_-,Q)\; =\; mathrm\{res\}(Q\_-,P)$ where $P\_-(z)\; =\; P(-z)$

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.

* In wavelet theory, the resultant is closely related to the determinant of the transfer matrix of a refinable function.

See also

Elimination theory

References

* [http://mathworld.wolfram.com/Resultant.html Weisstein, Eric W. "Resultant." From MathWorld--A Wolfram Web Resource.]