return the greatest common right divisor in the completely factored form
FactoredGCRD(U, V, x, case)
a completely factored Ore operator
an Ore operator
the name of the independent variable
a parameter indicating the case of the equation ('differential' or 'shift')
Given a completely factored Ore operator U and a non-factored Ore operator V, the LinearOperators[FactoredGCRD] function returns the greatest common right divisor (GCRD) in the completely factored form.
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.
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+x⁢D+x+1⁢D2+D3.
There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].
a ≔ FactoredOrePoly⁡−1,x,3,x
b ≔ OrePoly⁡0,0,2⁢x3+4⁢x2,x4
Abramov, S.A., and Zima, E.V. "Minimal Completely Factorable Annihilators." Proc. ISSAC'97. 1997.
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