Sqrfree - Maple Help

Sqrfree

inert square-free factorization function

 Calling Sequence Sqrfree(a)

Parameters

 a - multivariate polynomial or a multivariate rational function

Description

 • The Sqrfree function is a placeholder for representing the square-free factorization of the multivariate polynomial or rational function a over a unique factorization domain. It is used in conjunction with either mod, modp1 or evala which define the coefficient domain as described below.
 • The Sqrfree function returns a data structure of the form $\left[u,\left[\left[{f}_{1},{e}_{1}\right],\mathrm{...},\left[{f}_{n},{e}_{n}\right]\right]\right]$ such that $a=u{f}_{1}^{{e}_{1}}\cdots {f}_{n}^{{e}_{n}}$ and ${f}_{i}$ is primitive and square-free and $u$ is the leading coefficient of a. That is, $\mathrm{Gcd}\left({f}_{i},\frac{\partial }{\partial {x}_{j}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}_{i}\right)=1$ for all $i$ and $\mathrm{Gcd}\left({f}_{i},{f}_{j}\right)=1$ for $i\ne j$.
 • The call Sqrfree(a) mod p computes the square-free factorization of the polynomial a modulo p a prime integer. The multivariate polynomial a must have rational coefficients or coefficients from an algebraic extension of the integers modulo p.
 • The call modp1(Sqrfree(a), p) computes the square-free factorization of the polynomial a in the modp1 representation modulo p a prime integer.
 • The call evala(Sqrfree(a)) computes the square-free factorization of the polynomial or the rational function a where the coefficients of a are algebraic numbers (or functions) defined by RootOf or radicals. See evala,Sqrfree for more information.

Examples

 > $\mathrm{Sqrfree}\left(2{x}^{2}+6x+6\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}7$
 $\left[{2}{,}\left[\left[{{x}}^{{2}}{+}{3}{}{x}{+}{3}{,}{1}\right]\right]\right]$ (1)
 > $\mathrm{Sqrfree}\left(4{x}^{2}+4x+1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}7$
 $\left[{4}{,}\left[\left[{x}{+}{4}{,}{2}\right]\right]\right]$ (2)
 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{2}+x+1\right)\right)$
 ${\mathrm{\alpha }}$ (3)
 > $\mathrm{Sqrfree}\left(\mathrm{\alpha }{x}^{3}+\left(\mathrm{\alpha }+1\right){x}^{2}+x+\mathrm{\alpha }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 $\left[{\mathrm{\alpha }}{,}\left[\left[{x}{+}{\mathrm{\alpha }}{,}{3}\right]\right]\right]$ (4)
 > $\mathrm{Sqrfree}\left({x}^{2}+{y}^{2}+\mathrm{\alpha }+1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 $\left[{1}{,}\left[\left[{y}{+}{x}{+}{\mathrm{\alpha }}{,}{2}\right]\right]\right]$ (5)
 > $\mathrm{evala}\left(\mathrm{Sqrfree}\left({x}^{3}-2x-{2}^{\frac{1}{2}}{x}^{2}+2{2}^{\frac{1}{2}}\right)\right)$
 $\left[{1}{,}\left[\left[{x}{+}\sqrt{{2}}{,}{1}\right]{,}\left[{x}{-}\sqrt{{2}}{,}{2}\right]\right]\right]$ (6)
 > $\mathrm{evala}\left(\mathrm{Sqrfree}\left(3{x}^{2}+6\mathrm{RootOf}\left({x}^{2}-2\right)x+6\right)\right)$
 $\left[{3}{,}\left[\left[{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right){+}{x}{,}{2}\right]\right]\right]$ (7)