eigenring

Usage

eigenring L
eigenring A

Parameter	Type	Description
L	$\mathbbm{Q}[x,\frac d{dx}]$	A differential operator
A	$\mathbbm{Q}(x)^{n,n}$	A matrix of fractions

Description

eigenring(L) returns a basis $R_1,\dots,R_m$ of the eigenring of L, i.e.the set of operators $R\in \mathbbm{Q}(x)[\frac d{dx}]$ of order strictly less than the order of L and such that $L R = S L$ for some operator S.
eigenring(A) returns an $n$ by $nm$ matrix $M = [B_1\vert\dots\vert B_m]$ such that $B_1,\dots,B_m$ for a basis of the eigenring of $Y' = A Y$, i.e. the set of matrices $B\in \mathbbm{Q}(x)^{n,n}$ such that $B' = A B - B A$.

Example

We compute the eigenring of the differential system

$$Y' = \left [\begin {array}{ccc} {x}^{3}&{\frac {{x}^{5}+{x}^... ...{2}}}&{\frac {-3+{x}^{4}+3\,x}{{x}^{2}}}\end {array}\right ] Y$$ (3)

as follows:

1 --> A := system(x^3,(x^5+x^2-3+x^4)/x^2,(-3+x^4+3*x)/x^2,
                 -x^3,-(x^5-3+x^4)/x^2,-(-x^2-3+x^4+3*x)/x^2,
                  x^3,(x^5-3+x^4)/x^2,(-3+x^4+3*x)/x^2);
2 --> e := eigenring(A);
3 --> tex(e);

$$\pmatrix{ 1 & 0 & 0 & x^{2}+2\,x+1 & {{x^{5}+x^{4}+x^{2}+x+1... ... {{x^{5}-x^{3}-1} \over {x^{3}}} & {{x-1} \over {x^{3}}}\cr }$$

This means that a basis of the eigenring of (3) is

$$\pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr }, \quad... ...-1} \over {x^{3}}} & {{-x^{4}+x^{3}+x-1} \over {x^{3}}} \cr },$$

$$\mbox{and } \pmatrix{ x^{2}+x & {{x^{4}+x^{3}+x+1} \over {x^... ... {{x^{5}-x^{3}-1} \over {x^{3}}} & {{x-1} \over {x^{3}}}\cr }$$

Usage within MAPLE

• When using eigenring(A,D,x) from inside MAPLE, the matrix $[B_1\vert\dots\vert B_m]$ returned from BERNINA is transformed into the array of matrices $[B_1,\dots,B_m]$. So the above example in MAPLE would be:

  > A := matrix(3, 3, [x^3,(x^5+x^2-3+x^4)/x^2,(-3+x^4+3*x)/x^2,
                      -x^3,-(x^5-3+x^4)/x^2,-(-x^2-3+x^4+3*x)/x^2,
                       x^3,(x^5-3+x^4)/x^2,(-3+x^4+3*x)/x^2]);
  > eigenring(A, D, x);
  

  
  
  $$\left[ \pmatrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr }... ...r {x^{3}}} & {{-x^{4}+x^{3}+x-1} \over {x^{3}}} \cr }, \right.$$
  
  
  
  
  $$\left. \pmatrix{ x^{2}+x & {{x^{4}+x^{3}+x+1} \over {x^{2}}}... ...-x^{3}-1} \over {x^{3}}} & {{x-1} \over {x^{3}}}\cr } \right]$$