compute the closure of a linear differential operator
Closure(L, Dx, x, p, func, tord)
polynomial in Dx with coefficients that are polynomials in x
variable, denoting the differential operator w.r.t. x
(optional) irreducible polynomial in x
(optional) Maple command or user-defined procedure
(optional) equation of the form termorder=TO
Let L be a linear differential operator, given as a polynomial in Dx with univariate polynomial coefficients in x over a field k of characteristic zero. The command Closure(L,Dx,x) constructs a basis of the closure of L, whose elements R satisfy P·L=f·R for an operator P and polynomial f in k[x] not dividing P on the left.
If an optional fourth argument p is provided, Closure(L,Dx,x,p) constructs a local closure of L at the irreducible polynomial p. The output is a list of generators whose elements R satisfy P·L=p·R.
A function may be specified using the optional argument func. It is applied to the coefficients of the collected result. Often factor or expand will be used.
A Groebner basis computation with respect to a particular term ordering can be applied to the closure with the optional argument 'termorder'=TO where TO is of type MonomialOrder.
For the given differential operator L
L ≔ Dx4⁢x2−4⁢Dx3⁢x+6−x4−2⁢x3⁢Dx2+2⁢Dx⁢x2+x5+x4−2⁢x
compute the closure of L:
C ≔ Closure⁡L,Dx,x
In the following example, we apply the Groebner basis computation with term ordering plex⁡Dx,x to the computed differential closure.
A ≔ Ore_algebra:-diff_algebra⁡Dx,x,polynom=x:
TO ≔ Groebner:-MonomialOrder⁡A,'plex'⁡Dx,x:
Compute the local closure of L at p = x^2+1. Only one of the polynomials in C satisfies P·L=p·R.
Tsai, H. "Weyl closure of a linear differential operator." Journal of Symbolic Computation Vol. 29 No. 4-5 (2000): 747-775.
Chyzak, F.; Dumas, P.; Le, H.Q.; Martins, J.; Mishna, M.; Salvy, B. "Taming apparent singularities via Ore closure." In preparation.
The DEtools[Closure] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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