sqrfree - Maple Programming Help

sqrfree

square-free factorization function

 Calling Sequence sqrfree(a) sqrfree(a, x)

Parameters

 a - multivariate polynomial or rational function x - name or list or set of names

Description

 • The sqrfree function computes the square-free factorization of the multivariate polynomial or the rational function a in the variable(s) x over an algebraic number field.
 • The square-free factorization is returned in the form $\left[u,\left[\left[{f}_{1},{e}_{1}\right],\mathrm{...},\left[{f}_{n},{e}_{n}\right]\right]\right]$ such that $a=u{f}_{1}^{{e}_{1}}\cdots {f}_{n}^{{e}_{n}}$ where ${f}_{i}$ is primitive and square-free, that is, $\mathrm{gcd}\left({f}_{i},\frac{\partial }{\partial x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}_{i}\right)=1$ for all i and $\mathrm{gcd}\left({f}_{i},{f}_{j}\right)=1$ for all $i\ne j$ hence u is the content(a,x) times a unit.
 • In the case of two arguments, the second argument x denotes the main variable(s).  In the case of one argument, all the names in the polynomial a are used as variables.
 • Note that the ${e}_{i}$ are not necessarily distinct as partial factorizations in the input are preserved as much as possible.

Examples

 > $\mathrm{sqrfree}\left(-12{x}^{3}+9x+3\right)$
 $\left[{-3}{,}\left[\left[{x}{-}{1}{,}{1}\right]{,}\left[{1}{+}{2}{}{x}{,}{2}\right]\right]\right]$ (1)
 > $f≔{x}^{3}y-{x}^{3}-{x}^{2}{y}^{2}+{x}^{2}y$
 ${f}{≔}{{x}}^{{3}}{}{y}{-}{{x}}^{{2}}{}{{y}}^{{2}}{-}{{x}}^{{3}}{+}{{x}}^{{2}}{}{y}$ (2)
 > $\mathrm{sqrfree}\left(f,x\right)$
 $\left[{y}{-}{1}{,}\left[\left[{x}{,}{2}\right]{,}\left[{-}{y}{+}{x}{,}{1}\right]\right]\right]$ (3)
 > $\mathrm{sqrfree}\left(f,y\right)$
 $\left[{-}{{x}}^{{2}}{,}\left[\left[{-}{x}{}{y}{+}{{y}}^{{2}}{+}{x}{-}{y}{,}{1}\right]\right]\right]$ (4)
 > $\mathrm{sqrfree}\left(f,\left[x,y\right]\right)$
 $\left[{1}{,}\left[\left[{x}{,}{2}\right]{,}\left[{y}{-}{1}{,}{1}\right]{,}\left[{-}{y}{+}{x}{,}{1}\right]\right]\right]$ (5)
 > $g≔\left(x+y+1\right)\mathrm{expand}\left({\left(x+y+1\right)}^{2}\right){\left(x-y-3\right)}^{3}\left(3x+6y-21\right)$
 ${g}{≔}\left({x}{+}{y}{+}{1}\right){}\left({{x}}^{{2}}{+}{2}{}{x}{}{y}{+}{{y}}^{{2}}{+}{2}{}{x}{+}{2}{}{y}{+}{1}\right){}{\left({x}{-}{y}{-}{3}\right)}^{{3}}{}\left({3}{}{x}{+}{6}{}{y}{-}{21}\right)$ (6)
 > $\mathrm{sqrfree}\left(g,\left[x,y\right]\right)$
 $\left[{3}{,}\left[\left[{x}{+}{y}{+}{1}{,}{3}\right]{,}\left[{x}{-}{y}{-}{3}{,}{3}\right]{,}\left[{x}{+}{2}{}{y}{-}{7}{,}{1}\right]\right]\right]$ (7)
 > $h≔\frac{{\left(x+y\right)}^{3}}{\mathrm{expand}\left({\left({x}^{2}-{y}^{2}\right)}^{2}\left({x}^{2}+1\right)\right)}$
 ${h}{≔}\frac{{\left({x}{+}{y}\right)}^{{3}}}{{{x}}^{{6}}{-}{2}{}{{x}}^{{4}}{}{{y}}^{{2}}{+}{{x}}^{{2}}{}{{y}}^{{4}}{+}{{x}}^{{4}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}}$ (8)
 > $\mathrm{sqrfree}\left(h\right)$
 $\left[{1}{,}\left[\left[{x}{+}{y}{,}{1}\right]{,}\left[{{x}}^{{2}}{+}{1}{,}{-1}\right]{,}\left[{-}{y}{+}{x}{,}{-2}\right]\right]\right]$ (9)
 > $\mathrm{sqrfree}\left(h,y\right)$
 $\left[\frac{{1}}{{{x}}^{{2}}{+}{1}}{,}\left[\left[{x}{+}{y}{,}{1}\right]{,}\left[{y}{-}{x}{,}{-2}\right]\right]\right]$ (10)