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, $\gcd \left(f_i\,\frac{\partial }{\partial x}{f}_i\right)=1$ for all i and $\gcd \left(f_i\,f_j\right)=1$ 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.

$\mathrm{sqrfree}\left(-12x^3+9x+3\right)$

$\left[\mathrm{-3}\,\left[\left[x-1\,1\right]\,\left[1+2x\,2\right]\right]\right]$

$f≔x^{3}y-x^3-x^2{y}^2+x^{2}y$

$f≔x^{3}y-x^2{y}^2-x^3+x^{2}y$

$\mathrm{sqrfree}\left(f\,x\right)$

$\left[y-1\,\left[\left[x\,2\right]\,\left[-y+x\,1\right]\right]\right]$

$\mathrm{sqrfree}\left(f\,y\right)$

$\left[-x^2\,\left[\left[-xy+y^2+x-y\,1\right]\right]\right]$

$\mathrm{sqrfree}\left(f\,\left[x\,y\right]\right)$

$\left[1\,\left[\left[x\,2\right]\,\left[y-1\,1\right]\,\left[-y+x\,1\right]\right]\right]$

$g≔\left(x+y+1\right)\mathrm{expand}\left({\left(x+y+1\right)}^2\right){\left(x-y-3\right)}^3\left(3x+6y-21\right)$

$g≔\left(x+y+1\right)\left(x^2+2xy+y^2+2x+2y+1\right){\left(x-y-3\right)}^3\left(3x+6y-21\right)$

$\mathrm{sqrfree}\left(g\,\left[x\,y\right]\right)$

$\left[3\,\left[\left[x+y+1\,3\right]\,\left[x-y-3\,3\right]\,\left[x+2y-7\,1\right]\right]\right]$

$h≔\frac{{\left(x+y\right)}^3}{\mathrm{expand}\left({\left(x^2-y^2\right)}^2\left(x^2+1\right)\right)}$

$h≔\frac{{\left(x+y\right)}^3}{x^6-2x^4{y}^2+x^2{y}^4+x^4-2x^2{y}^2+y^4}$

$\mathrm{sqrfree}\left(h\right)$

$\left[1\,\left[\left[x+y\,1\right]\,\left[x^2+1\,\mathrm{-1}\right]\,\left[-y+x\,\mathrm{-2}\right]\right]\right]$

$\mathrm{sqrfree}\left(h\,y\right)$

$\left[\frac{1}{x^2+1}\,\left[\left[x+y\,1\right]\,\left[y-x\,\mathrm{-2}\right]\right]\right]$