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PolynomialTools

 GreatestFactorialFactorization
 compute a greatest factorial factorization of a univariate polynomial

 Calling Sequence GreatestFactorialFactorization(f,x)

Parameters

 f - polynomial in x x - indeterminate

Description

 • The GreatestFactorialFactorization command computes a greatest factorial factorization $[c,[[\mathrm{g1},\mathrm{e1}],[\mathrm{g2},\mathrm{e2}],...]]$ of f w.r.t. x. It satisfies the following properties.
 $f=c\mathrm{g1}\left(x\right)\mathrm{g1}\left(x-1\right)...\mathrm{g1}\left(x-\mathrm{e1}+1\right)\mathrm{g2}\left(x\right)\mathrm{g2}\left(x-1\right)...\mathrm{g2}\left(x-\mathrm{e2}+1\right)...$
 $\mathrm{gcd}\left(\mathrm{gi}\left(x\right)\mathrm{gi}\left(x-1\right)...\mathrm{gi}\left(x-\mathrm{e1}+1\right),\mathrm{gj}\left(x+1\right)\mathrm{gj}\left(x+\mathrm{ej}\right)\right)=1$ for $1<=i<=j$
 $c$ is constant w.r.t. x, and $\mathrm{g1},...$ are nonconstant primitive polynomials w.r.t. x, and $0<\mathrm{e1}<\mathrm{e2}<\mathrm{...}$ are integers.
 • The greatest factorial factorization is unique up to multiplication by units.
 • GreatestFactorialFactorization can handle the same types of coefficients as the Maple function gcd.
 • If f is constant w.r.t. x, then the return value is $\left[f,\left[\right]\right]$.
 • Partial factorizations of the input are not taken into account.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialTools}\right):$
 > $\mathrm{GreatestFactorialFactorization}\left(-{x}^{8}+{x}^{2},x\right)$
 $\left[{-}{1}{,}\left[\left[{x}{,}{1}\right]{,}\left[{{x}}^{{2}}{+}{x}{+}{1}{,}{2}\right]{,}\left[{x}{+}{1}{,}{3}\right]\right]\right]$ (1)
 > $\mathrm{GreatestFactorialFactorization}\left(\mathrm{expand}\left(\mathrm{pochhammer}\left(x,3\right)\mathrm{pochhammer}\left(x,5\right)\right),x\right)$
 $\left[{1}{,}\left[\left[{x}{+}{2}{,}{3}\right]{,}\left[{x}{+}{4}{,}{5}\right]\right]\right]$ (2)

References

 Paule, Peter. "Greatest factorial factorization and symbolic summation." Journal of Symbolic Computation Vol. 20, (1995): 235-268.
 Gerhard, Juergen. "Modular algorithms for polynomial basis conversion and greatest factorial factorization." Proceedings of the Seventh Rhine Workshop on Computer Algebra, RWCA pp. 125-141 ed. T. Mulders, 2000.