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DEtools

 Desingularize
 desingularize a linear differential operator

 Calling Sequence Desingularize(L, Dx, x, func)

Parameters

 L - polynomial in Dx with coefficients that are polynomials in x Dx - variable, denoting the differential operator w.r.t. x x - variable func - (optional) procedure

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 Desingularize(L,Dx,x) constructs a linear differential operator R such that any solution of $L\left(y\right)=0$ is also a solution of $R\left(y\right)=0$ and R has no apparent singularities.  The operator R is said to maximally desingularize L, and will be right divisible by L over the field $k\left(x\right)$.
 • An apparent singularity is a point $p$ where the leading coefficient of L vanishes, yet $p$ is not a pole of any holomorphic solution of $L\left(y\right)=0$. In this case there will exist $d$ linearly independent solutions at $p$ where $d$ is the order of L.
 • A function may be specified using the optional argument func. It is applied to the coefficients of the collected result. Often simplify or factor will be used.

Examples

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

For the given differential operator L

 > $L≔\left(24{x}^{3}-18{x}^{4}+{x}^{8}+6{x}^{5}-{x}^{6}\right){\mathrm{Dx}}^{7}+\left(6{x}^{5}+72{x}^{3}-30{x}^{4}-8{x}^{7}-72{x}^{2}\right){\mathrm{Dx}}^{6}+\left(-144{x}^{2}+36{x}^{6}+72{x}^{3}-2{x}^{7}+144x-18{x}^{4}\right){\mathrm{Dx}}^{5}+\left(24{x}^{3}+36{x}^{6}+144x-144-72{x}^{2}-120{x}^{5}-8{x}^{7}-{x}^{10}+{x}^{8}\right){\mathrm{Dx}}^{4}+\left(-24{x}^{5}-{x}^{10}-6{x}^{7}+{x}^{8}+18{x}^{6}\right){\mathrm{Dx}}^{3}+\left(36{x}^{5}-6{x}^{6}-72{x}^{4}+2{x}^{9}\right){\mathrm{Dx}}^{2}+\left(-36{x}^{4}+12{x}^{5}-10{x}^{8}+2{x}^{9}\right)\mathrm{Dx}+64{x}^{7}-12{x}^{4}-32{x}^{8}+8{x}^{9}+{x}^{12}-{x}^{10}$
 ${L}{≔}\left({{x}}^{{8}}{-}{{x}}^{{6}}{+}{6}{}{{x}}^{{5}}{-}{18}{}{{x}}^{{4}}{+}{24}{}{{x}}^{{3}}\right){}{{\mathrm{Dx}}}^{{7}}{+}\left({-}{8}{}{{x}}^{{7}}{+}{6}{}{{x}}^{{5}}{-}{30}{}{{x}}^{{4}}{+}{72}{}{{x}}^{{3}}{-}{72}{}{{x}}^{{2}}\right){}{{\mathrm{Dx}}}^{{6}}{+}\left({-}{2}{}{{x}}^{{7}}{+}{36}{}{{x}}^{{6}}{-}{18}{}{{x}}^{{4}}{+}{72}{}{{x}}^{{3}}{-}{144}{}{{x}}^{{2}}{+}{144}{}{x}\right){}{{\mathrm{Dx}}}^{{5}}{+}\left({-}{{x}}^{{10}}{+}{{x}}^{{8}}{-}{8}{}{{x}}^{{7}}{+}{36}{}{{x}}^{{6}}{-}{120}{}{{x}}^{{5}}{+}{24}{}{{x}}^{{3}}{-}{72}{}{{x}}^{{2}}{+}{144}{}{x}{-}{144}\right){}{{\mathrm{Dx}}}^{{4}}{+}\left({-}{{x}}^{{10}}{+}{{x}}^{{8}}{-}{6}{}{{x}}^{{7}}{+}{18}{}{{x}}^{{6}}{-}{24}{}{{x}}^{{5}}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({2}{}{{x}}^{{9}}{-}{6}{}{{x}}^{{6}}{+}{36}{}{{x}}^{{5}}{-}{72}{}{{x}}^{{4}}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({2}{}{{x}}^{{9}}{-}{10}{}{{x}}^{{8}}{+}{12}{}{{x}}^{{5}}{-}{36}{}{{x}}^{{4}}\right){}{\mathrm{Dx}}{+}{64}{}{{x}}^{{7}}{-}{12}{}{{x}}^{{4}}{-}{32}{}{{x}}^{{8}}{+}{8}{}{{x}}^{{9}}{+}{{x}}^{{12}}{-}{{x}}^{{10}}$ (1)

compute a desingularizing operator for L:

 > $M≔\mathrm{Desingularize}\left(L,\mathrm{Dx},x,\mathrm{factor}\right)$
 ${M}{≔}{1728252}{}{{\mathrm{Dx}}}^{{8}}{+}\left({54154}{}{{x}}^{{7}}{+}{161694}{}{{x}}^{{6}}{+}{263753}{}{{x}}^{{5}}{+}{452649}{}{{x}}^{{4}}{-}{324882}{}{{x}}^{{3}}{+}{1728252}\right){}{{\mathrm{Dx}}}^{{7}}{+}\left({-}{433232}{}{{x}}^{{6}}{-}{1293552}{}{{x}}^{{5}}{-}{2218332}{}{{x}}^{{4}}{-}{2969808}{}{{x}}^{{3}}{+}{974646}{}{{x}}^{{2}}{+}{1728252}\right){}{{\mathrm{Dx}}}^{{6}}{+}\left({-}{108308}{}{{x}}^{{6}}{+}{1626156}{}{{x}}^{{5}}{+}{5185170}{}{{x}}^{{4}}{+}{9891042}{}{{x}}^{{3}}{+}{9684162}{}{{x}}^{{2}}{-}{1949292}{}{x}{+}{1728252}\right){}{{\mathrm{Dx}}}^{{5}}{+}\left({-}{54154}{}{{x}}^{{9}}{-}{161694}{}{{x}}^{{8}}{-}{263753}{}{{x}}^{{7}}{-}{560957}{}{{x}}^{{6}}{+}{976266}{}{{x}}^{{5}}{-}{2924106}{}{{x}}^{{4}}{-}{10379136}{}{{x}}^{{3}}{-}{27461604}{}{{x}}^{{2}}{-}{45113328}{}{x}{+}{3677544}\right){}{{\mathrm{Dx}}}^{{4}}{-}{x}{}\left({54154}{}{{x}}^{{8}}{+}{161694}{}{{x}}^{{7}}{+}{263753}{}{{x}}^{{6}}{+}{452649}{}{{x}}^{{5}}{-}{324882}{}{{x}}^{{4}}{+}{1728252}{}{x}{+}{13826016}\right){}{{\mathrm{Dx}}}^{{3}}{+}\left({108308}{}{{x}}^{{8}}{+}{323388}{}{{x}}^{{7}}{+}{635814}{}{{x}}^{{6}}{+}{253914}{}{{x}}^{{5}}{+}{974646}{}{{x}}^{{4}}{-}{1728252}{}{{x}}^{{2}}{-}{10369512}{}{x}{-}{20739024}\right){}{{\mathrm{Dx}}}^{{2}}{+}\left({108308}{}{{x}}^{{8}}{-}{218152}{}{{x}}^{{7}}{-}{981126}{}{{x}}^{{6}}{-}{2600232}{}{{x}}^{{5}}{-}{520824}{}{{x}}^{{4}}{-}{1728252}{}{{x}}^{{2}}{-}{6913008}{}{x}{-}{10369512}\right){}{\mathrm{Dx}}{+}{54154}{}{{x}}^{{11}}{+}{161694}{}{{x}}^{{10}}{+}{263753}{}{{x}}^{{9}}{+}{560957}{}{{x}}^{{8}}{-}{759650}{}{{x}}^{{7}}{+}{538258}{}{{x}}^{{6}}{+}{2595900}{}{{x}}^{{5}}{+}{10387932}{}{{x}}^{{4}}{+}{31684620}{}{{x}}^{{3}}{-}{1728252}{}{{x}}^{{2}}{-}{3456504}{}{x}{-}{3456504}$ (2)
 > $Q,R≔\mathrm{op}\left(\mathrm{DEtools}\left['\mathrm{rightdivision}'\right]\left(M,L,\left[\mathrm{Dx},x\right]\right)\right):$

Hence, $R=Q·L+R$ where

 > $Q$
 $\frac{{1728252}{}{\mathrm{Dx}}}{{{x}}^{{3}}{}\left({{x}}^{{5}}{-}{{x}}^{{3}}{+}{6}{}{{x}}^{{2}}{-}{18}{}{x}{+}{24}\right)}{+}\frac{{54154}{}{{x}}^{{7}}{+}{161694}{}{{x}}^{{6}}{+}{263753}{}{{x}}^{{5}}{+}{452649}{}{{x}}^{{4}}{-}{324882}{}{{x}}^{{3}}{+}{1728252}}{{{x}}^{{3}}{}\left({{x}}^{{5}}{-}{{x}}^{{3}}{+}{6}{}{{x}}^{{2}}{-}{18}{}{x}{+}{24}\right)}$ (3)
 > $R$
 ${0}$ (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[Desingularize] command was introduced in Maple 15.