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DEtools

 Closure
 compute the closure of a linear differential operator

 Calling Sequence Closure(L, Dx, x, p, func, tord)

Parameters

 L - polynomial in Dx with coefficients that are polynomials in x Dx - variable, denoting the differential operator w.r.t. x x - variable p - (optional) irreducible polynomial in x func - (optional) Maple command or user-defined procedure tord - (optional) equation of the form termorder=TO

Description

 • 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 $P\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}L=f\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}R$ 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 $P\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}L=p\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}R$.
 • 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.

Examples

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

For the given differential operator L

 > $L≔{\mathrm{Dx}}^{4}{x}^{2}-4{\mathrm{Dx}}^{3}x+\left(6-{x}^{4}-2{x}^{3}\right){\mathrm{Dx}}^{2}+2\mathrm{Dx}{x}^{2}+{x}^{5}+{x}^{4}-2x$
 ${L}{≔}{{\mathrm{Dx}}}^{{4}}{}{{x}}^{{2}}{-}{4}{}{{\mathrm{Dx}}}^{{3}}{}{x}{+}\left({-}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{+}{6}\right){}{{\mathrm{Dx}}}^{{2}}{+}{2}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{{x}}^{{5}}{+}{{x}}^{{4}}{-}{2}{}{x}$ (1)

compute the closure of L:

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

In the following example, we apply the Groebner basis computation with term ordering $\mathrm{plex}\left(\mathrm{Dx},x\right)$ to the computed differential closure.

 > $A≔\mathrm{Ore_algebra}:-\mathrm{diff_algebra}\left(\left[\mathrm{Dx},x\right],\mathrm{polynom}=x\right):$
 > $\mathrm{TO}≔\mathrm{Groebner}:-\mathrm{MonomialOrder}\left(A,'\mathrm{plex}'\left(\mathrm{Dx},x\right)\right):$
 > $\mathrm{Closure}\left(L,\mathrm{Dx},x,'\mathrm{termorder}'=\mathrm{TO}\right)$
 $\left[{{\mathrm{Dx}}}^{{4}}{}{{x}}^{{2}}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{{x}}^{{5}}{-}{4}{}{{\mathrm{Dx}}}^{{3}}{}{x}{+}{{x}}^{{4}}{+}{2}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{6}{}{{\mathrm{Dx}}}^{{2}}{-}{2}{}{x}{,}{{\mathrm{Dx}}}^{{6}}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{4}}{-}{{\mathrm{Dx}}}^{{4}}{}{x}{-}{2}{}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{3}}{+}{{x}}^{{5}}{-}{12}{}{{\mathrm{Dx}}}^{{3}}{}{x}{-}{{\mathrm{Dx}}}^{{2}}{}{{x}}^{{2}}{+}{2}{}{{x}}^{{4}}{-}{14}{}{{\mathrm{Dx}}}^{{3}}{+}{12}{}{\mathrm{Dx}}{}{{x}}^{{2}}{+}{{x}}^{{3}}{-}{6}{}{{\mathrm{Dx}}}^{{2}}{+}{10}{}{\mathrm{Dx}}{}{x}{+}{18}{}{x}{+}{10}\right]$ (3)

Compute the local closure of L at p = x^2+1.  Only one of the polynomials in C satisfies $P\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}L=p\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}R$.

 > $\mathrm{Closure}\left(L,\mathrm{Dx},x,{x}^{2}+1\right)$
 $\left[{{x}}^{{2}}{}{{\mathrm{Dx}}}^{{4}}{-}{4}{}{x}{}{{\mathrm{Dx}}}^{{3}}{+}\left({-}{{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{+}{6}\right){}{{\mathrm{Dx}}}^{{2}}{+}{2}{}{{x}}^{{2}}{}{\mathrm{Dx}}{+}{{x}}^{{5}}{+}{{x}}^{{4}}{-}{2}{}{x}\right]$ (4)

References

 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.

Compatibility

 • The DEtools[Closure] command was introduced in Maple 15.