construct completely factorable annihilator
FactoredAnnihilator(expr, x, case)
an algebraic expression
the name of the independent variable
a parameter indicating the case of the equation ('differential' or 'shift')
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 −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 −1+xD⁡x⁡D⁢x2+4⁡D.
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.
expr ≔ ⅇxx−1+x2+1
L ≔ LinearOperators[FactoredAnnihilator]⁡expr,x,'differential'
L ≔ FactoredOrePoly⁡x4+6⁢x2−2⁢x+3x3−x2+x+1⁢x2+1⁢x−1,1,1x−12,1
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.
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