Given an algebraic expression expr, the LinearOperators[FactoredAnnihilator] function returns a completely factored Ore operator that is an annihilator for expr. That is, applying this operator to expr yields zero. If such an operator does not exist, the function returns $\mathrm{-1}$.

A completely factored Ore operator is an operator that can be factored into a product of factors of degree at most one.

A completely factored Ore operator is represented by a structure that consists of the keyword FactoredOrePoly and a sequence of lists. Each list consists of two elements and describes a first degree factor. The first element provides the zero degree coefficient and the second element provides the first degree coefficient. For example, in the differential case with a differential operator D, FactoredOrePoly([-1, x], [x, 0], [4, x^2], [0, 1]) describes the operator $\left(-1+\mathrm{xD}\right)\left(x\right)\left(x^2\mathrm{D}+4\right)\left(\mathrm{D}\right)$.

There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

To get the completely factorable annihilator, the expression expr must be a d'Alembertian term. The main property of a d'Alembertian term is that it is annihilated by a linear operator that can be written as a composition of operators of the first degree. The set of d'Alembertian terms has a ring structure. The package recognizes some basic d'Alembertian terms and their ring-operation closure terms. The result of the substitution of a rational term for the independent variable in the d'Alembertian term is also a d'Alembertian term.

$\mathrm{expr}≔\exp \left(\frac{x}{x-1}\right)+\mathrm{sqrt}\left(x^2+1\right)$

$\mathrm{expr}≔{\ⅇ}^{\frac{x}{x-1}}+\sqrt{x^2+1}$

$L≔\mathrm{LinearOperators}\left[\mathrm{FactoredAnnihilator}\right]\left(\mathrm{expr}\,x\,'\mathrm{differential}'\right)$

$L≔\mathrm{FactoredOrePoly}\left(\left[\frac{x^4+6x^2-2x+3}{\left(x^3-x^2+x+1\right)\left(x^2+1\right)\left(x-1\right)}\,1\right]\,\left[\frac{1}{{\left(x-1\right)}^2}\,1\right]\right)$

$\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(L\,\mathrm{expr}\,x\,'\mathrm{differential}'\right)$

$0$

Abramov, S. A., and Zima, E. V. "Minimal Completely Factorable Annihilators." In Proceedings of ISSAC '97, pp. 290-297. Edited by Wolfgang Kuchlin. New York: ACM Press, 1997.