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, $\mathrm{Gcd}\left(f_i\,\frac{\partial }{\partial x_j}{f}_i\right)=1$ 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.

$\mathrm{Sqrfree}\left(2x^2+6x+6\right)\mathbf{mod}7$

$\left[2\,\left[\left[x^2+3x+3\,1\right]\right]\right]$

$\mathrm{Sqrfree}\left(4x^2+4x+1\right)\mathbf{mod}7$

$\left[4\,\left[\left[x+4\,2\right]\right]\right]$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(x^2+x+1\right)\right)$

$\mathrm{\alpha }$

$\mathrm{Sqrfree}\left(\mathrm{\alpha }{x}^3+\left(\mathrm{\alpha }+1\right)x^2+x+\mathrm{\alpha }\right)\mathbf{mod}2$

$\left[\mathrm{\alpha }\,\left[\left[x+\mathrm{\alpha }\,3\right]\right]\right]$

$\mathrm{Sqrfree}\left(x^2+y^2+\mathrm{\alpha }+1\right)\mathbf{mod}2$

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

$\mathrm{evala}\left(\mathrm{Sqrfree}\left(x^3-2x-2^{\frac{1}{2}}{x}^2+22^{\frac{1}{2}}\right)\right)$

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

$\mathrm{evala}\left(\mathrm{Sqrfree}\left(3x^2+6\mathrm{RootOf}\left(x^2-2\right)x+6\right)\right)$

$\left[3\,\left[\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\right)+x\,2\right]\right]\right]$