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

$\backslash mathrm\{res\}(P,Q)\; =\; \backslash prod\_\{P(x)=0\}\; \backslash prod\_\{Q(y)=0\}\; (y-x),\backslash ,$

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

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

Computation

• The resultant is the determinant of the Sylvester matrix.

• The above product can be rewritten to

$\backslash mathrm\{res\}(P,Q)\; =\; \backslash prod\_\{Q(y)=0\}\; P(y)\backslash ,$

and this expression remains unchanged if $P$ is reduced modulo $Q$.

• Let $P\text{'}\; =\; P\; \backslash mod\; Q$. 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 $\backslash prod\_\{Q(y)=0\}\; P\text{'}(y)\backslash ,$ 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.

$\backslash mathrm\{res\}(P,Q)\; =\; q^\{\backslash deg\; P\; -\; \backslash deg\; P\text{'}\}\; \backslash cdot\; \backslash mathrm\{res\}(P\text{'},Q)$

Continuing this procedure ends up in a variant of the Euclidean algorithm. This procedure needs quadratic runtime.

Properties

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

then

• $\backslash mathrm\{res\}(P\_-,Q)\; =\; \backslash mathrm\{res\}(Q\_-,P)$ where $P\_-(z)\; =\; P(-z)$

Applications

• 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 $\backslash mathbb\{A\}^2\_k$. 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.

References

• Mathworld entry

Polynomials Resultante