The GcdFreeBasis command uses repeated gcd computations to compute a gcd free basis B of the polynomials in S. B satisfies the following properties.

Each polynomial in S can be written as a constant times a product of polynomials from B.

The polynomials in B are pairwise coprime, that is, their gcd is constant.

With respect to cardinality, B is minimal with these properties. (This is equivalent to saying that B can be computed using gcds and divisions only, but not factorization.)

The gcd free basis is unique up to ordering and multiplication of the basis elements by constants.

GcdFreeBasis can handle the same types of coefficients as the Maple function gcd.

If S is a set, then the output is a set as well. If S is a list, then the output is also a list. The ordering of the elements in the result is not determined in either case.

Zero polynomials and constants are ignored. In particular, for a constant const, GcdFreeBasis([const]) returns [ ]. The empty set or list is a valid input and is returned unchanged.

A polynomial $f$ is considered constant by GcdFreeBasis if $\mathrm{degree}\left(f\right)$ returns $0$.

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\:$

$\mathrm{GcdFreeBasis}\left(\left[abcd\,abce\,abde\right]\right)$

$\left[c\,d\,e\,ab\right]$

$\mathrm{GcdFreeBasis}\left(\left\{x^6-1\,x^{10}-1\,x^{15}-1\right\}\right)$

$\left\{x-1\,x+1\,x^2-x+1\,x^2+x+1\,x^4-x^3+x^2-x+1\,x^4+x^3+x^2+x+1\,x^8-x^7+x^5-x^4+x^3-x+1\right\}$

Bach, Eric.; Driscoll, James.; and Shallit, Jeffrey. "Factor Refinement." Journal of Algorithms Vol. 15(2), (1993): 199-222.