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LinearOperators[MinimalAnnihilator] - construct the minimal annihilator

Calling Sequence

MinimalAnnihilator(L, expr, x, case)




completely factored Ore operator



Maple expression



name of the independent variable



parameter indicating the case of the equation ('differential' or 'shift')



Given a factored Ore operator L that is an annihilator for the expression expr, the LinearOperators[MinimalAnnihilator] function returns the minimal annihilator in non-factored form for expr.


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+xDxDx2+4D.


An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2x+xD+x+1D2+D3.


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


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.


Note: The operator L must annihilate expr, that is, satisfy L(expr)=0.














See Also

LinearOperators, LinearOperators[Apply], LinearOperators[converters], LinearOperators[FactoredAnnihilator]



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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