Loewy

Usage

Loewy L

Parameter	Type	Description
L	${\mathbb{Q}}[n,E]$	A difference operator of order at most 3

Returns

Returns either L, in which case L is completely reducible, or a right Loewy decomposition $[L_1,\dots,L_t]$ where $L = L_1 \cdots L_t$ and each $L_i \in {\mathbb{Q}}(n)[E]$ is a completely reducible right-factor of maximal order of $L_1 \cdots L_{i-1}$.

Example

We compute the Loewy decomposition of the difference equation

$$y(n+3) - n y(n+2) + (3n^2+9n+6)y(n+1) -(3n^3+3n^2)y(n) = 0$$ (4)

as follows:

1 --> L := E^3-n*E^2+(3*n^2+9*n+6)*E-3*n^3-3*n^2;
2 --> v := Loewy(L);
3 --> tex(v);

$$\left[ E-n , E^{2}+3\,n^{2}+3\,n \right]$$

This means that the operator of (4) factors as

$${\left({E-n}\right)}{\left({E^{2}+3\,n^{2}+3\,n}\right)}$$

where the second-order factor is a completely reducible right-factor of maximal order. Note that is is reducible in this example:

4 --> w := decompose(element(v,2));
5 --> tex(w);

$$\left[ n^{2}+3 , -\left(n\,E\right) , 1 \right]$$

which means that $E^{2}+3\,n^{2}+3\,n$ is a least common left multiple of $E - n \sqrt{-3}$ and $E + n \sqrt{-3}$.

See Also

decompose,factor