EigenRing

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\centerline{\fbox{\parbox{7.5cm}{\Large\bf UserFunction EigenRing} }}

\vspace*{5mm}\noindent{\bf EigenRing}$(u:LODO(C,v))$
The single argument $u$ has the type $LODO$ which abbreviates
{\em LinearDifferentialOperator}. The meaning of its parameters is as follows.

$C$: Coefficient type.

$x$: Differentiation variable.

\vspace*{1mm}\noindent{\bf Specification.}
Generators for the Eigenring are returned.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax, {\tt D(x,k)} means $\fracsm{d^k}{dx^k}$. The output for each minisession is returned in a separate window, $D\equiv \fracsm{d}{dx}$ and
$D^k\equiv \fracsm{d^k}{dx^k}$. The first example is taken from the article by M.~Singer.

\5{\tt z:=D(x,4);}

\5{\tt EigenRing z;}

\vspace*{3mm}

\vspace*{1mm}\hspace*{10mm}\fbox{\parbox{14cm}{

The Differential Operator: $D^4$

\vspace*{1mm}
The Eigenring is Generated by:

\vspace*{2mm}
$\{x^{3}D^3,x^{2}D^3,xD^3,D^3,x^{4}D^3+x^{3}D^2,$
\vspace*{2mm}

$\hspace*{1mm}x^{2}D^2,xD^2,D^2,\Frac{11}{10}x^{5}D^3+x^{4}D^2+x^{3}D,\Frac{1}{2}x^{4}D^3+x^{2}D,xD,D,$

\vspace*{2mm}
$\hspace*{2mm}\Frac{29}{30}x^{6}D^3+ \Frac{11}{10}x^{5}D^2+x^{4}D+x^{3},\Frac{7}{15}x^{5}D^3
+ \Frac{1}{2}x^{4}D^2+x^{2},\Frac{1}{6}x^{4}D^3+x,+1\}$  }}

\vspace*{5mm}

The next example is a completely reducible second-order operator.

\5{\tt z:=D(x,2)+2/x*D(x)-1;}

\5{\tt EigenRing z;}


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\fbox{\parbox{12cm}{

The Differential Operator:
$D^2+\fracsm{2}{x}D-1$

\vspace*{2mm}
The Eigenring is Generated by:

\vspace*{2mm}
$\{D+\fracsm{1}{x},+1\}$ }}

 

{\bf References}

\"{O}ystein Ore, {\em Formale Theorie der linearen Differentialgleichungen}, Erster Teil, Journal f\"{u}r die reine und angewandte Mathematik, vol. {\bf 167}, 221-234(1932); zweiter Teil, vol. {\bf 168}, 233-252(1932).

M.~Singer, {\em Testing Reducibility of Linear Differential Operators},
AAECC {\bf 7}, 77-104(1996).

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