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.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$L≔{\mathrm{Dx}}^4{x}^2-4{\mathrm{Dx}}^{3}x+\left(6-x^4-2x^3\right){\mathrm{Dx}}^2+2\mathrm{Dx}{x}^2+x^5+x^4-2x$

$L≔{\mathrm{Dx}}^4{x}^2-4{\mathrm{Dx}}^{3}x+\left(-x^4-2x^3+6\right){\mathrm{Dx}}^2+2\mathrm{Dx}{x}^2+x^5+x^4-2x$

$C≔\mathrm{Closure}\left(L\,\mathrm{Dx}\,x\right)$

$C≔\left[x^2{\mathrm{Dx}}^4-4x{\mathrm{Dx}}^3+\left(-x^4-2x^3+6\right){\mathrm{Dx}}^2+2x^2\mathrm{Dx}+x^5+x^4-2x\,x{\mathrm{Dx}}^6-x^2\left(x+2\right){\mathrm{Dx}}^4-2x\left(4x+5\right){\mathrm{Dx}}^3+\left(x^4+x^3-12x-6\right){\mathrm{Dx}}^2+2x^2\left(5x+4\right)\mathrm{Dx}+4x\left(5x+3\right)\,x{\mathrm{Dx}}^7+2{\mathrm{Dx}}^6-x^2\left(x+2\right){\mathrm{Dx}}^5-3x\left(4x+5\right){\mathrm{Dx}}^4+\left(x^4+x^3-36x-30\right){\mathrm{Dx}}^3+\left(14x^3+10x^2-24\right){\mathrm{Dx}}^2+2x\left(30x+19\right)\mathrm{Dx}+x^4+x^3+60x+22\right]$

$A≔\mathrm{Ore\_algebra}:-\mathrm{diff\_algebra}\left(\left[\mathrm{Dx}\,x\right]\,\mathrm{polynom}=x\right)\:$

$\mathrm{TO}≔\mathrm{Groebner}:-\mathrm{MonomialOrder}\left(A\,'\mathrm{plex}'\left(\mathrm{Dx}\,x\right)\right)\:$

$\mathrm{Closure}\left(L\,\mathrm{Dx}\,x\,'\mathrm{termorder}'=\mathrm{TO}\right)$

$\left[x^2{\mathrm{Dx}}^4-x^4{\mathrm{Dx}}^2-2x^3{\mathrm{Dx}}^2+x^5-4x{\mathrm{Dx}}^3+x^4+2x^2\mathrm{Dx}+6{\mathrm{Dx}}^2-2x\,{\mathrm{Dx}}^6-x^4{\mathrm{Dx}}^2-x{\mathrm{Dx}}^4-2x^3{\mathrm{Dx}}^2+x^5-12x{\mathrm{Dx}}^3-x^2{\mathrm{Dx}}^2+2x^4-14{\mathrm{Dx}}^3+12x^2\mathrm{Dx}+x^3-6{\mathrm{Dx}}^2+10x\mathrm{Dx}+18x+10\right]$

$\mathrm{Closure}\left(L\,\mathrm{Dx}\,x\,x^2+1\right)$

$\left[x^2{\mathrm{Dx}}^4-4x{\mathrm{Dx}}^3+\left(-x^4-2x^3+6\right){\mathrm{Dx}}^2+2x^2\mathrm{Dx}+x^5+x^4-2x\right]$

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.