factor

Usage

factor L

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

Description

Returns one of the following possible results:

(i)

L, in which case L is irreducible in ${\overline {\mathbb{Q}}}(n)[E]$.

(ii)

$[m,L_1,\dots,L_t]$ where $m \ge 0$ and $L_i \in {\mathbb{Q}}(n)[E]$, in which case each $L_t$ is irreducible in ${\overline {\mathbb{Q}}}(n)[E]$, and L can be expressed in terms of the $L_i$'s depending on $m$ as in the following table:

m	L
0	LeftLcm($L_1,\dots,L_t$)
1	$L_1 \cdots L_t$
2	LeftLcm($L_1, L_2 L_3$)
3	$L_1$ LeftLcm($L_2, L_3$)
4	$L_1$ LeftLcm($L_2, L_3$) $L_4$

(iii)

$[m,h(n),a(n,E),b(n,E),f_1(n,E),\dots,f_t(n,E)]$, where $m < 0$, $h(n) \in {\mathbb{Q}}[n]$ is irreducible and $a(n,E)$, $b(n,E)$, $f_1(n,E),\dots,f_t(n,E) \in {\mathbb{Q}}(n)[E]$, in which case each $f_i(n,E)$ is irreducible in ${\overline {\mathbb{Q}}}(n)[E]$, and L can be expressed depending on $m$ as in the following table:

m	L
-1	LeftLcm( $E + a(\alpha, n)/b(\alpha, n), f_1(n,E),\dots,f_t(n,E)$)
-4	$f_1(n,E)$ LeftLcm( $E + a(\alpha, n) / b(\alpha, n)$)
-5	$f_1(n,E)$ LeftLcm( $E + a(\alpha, n) / b(\alpha, n)$) $f_2(n,E)$

(iii)

$[m,h(n),a(n,E),b(n,E),f_1(n,E),\dots,f_t(n,E)]$, where $m < 0$, $h(n) \in {\mathbb{Q}}[n]$ is irreducible and $a(n,E)$, $b(n,E)$, $f_1(n,E),\dots,f_t(n,E) \in {\mathbb{Q}}(n)[E]$, in which case each $f_i(n,D)$ is irreducible in ${\overline {\mathbb{Q}}}(n)[E]$, and L can be expressed depending on $m$ as in the following table:

m	L
-1	LeftLcm( $E + a(\alpha, n)/b(\alpha, n), f_1(n,E),\dots,f_t(n,E)$)
-4	$f_1(n,E)$ LeftLcm( $E + a(\alpha, n) / b(\alpha, n)$)
-5	$f_1(n,E)$ LeftLcm( $E + a(\alpha, n) / b(\alpha, n)$) $f_2(n,E)$

where $\alpha$ ranges over all the roots of $h(n)$.

Example

We factor the difference equation

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

as follows:

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

$$\left[ -5 , n^{2}+3 , -\left(n\,E\right) , 1 , 1 , E-n \right]$$

This means that the operator of (3) can be written in the form

$$L = \mbox{LCLM}(E - \sqrt{-3} n, E + \sqrt{-3} n) (E - n)$$

Usage within MAPLE

In order not to conflict with the factor function in MAPLE, factor is available under MAPLE under the name efactor. In addition, when using efactor from inside MAPLE, the result returned from SHASTA is further transformed as in the case of the decompose function. So the above example in MAPLE would be:

> L := E^3+(-n-2)*E^2+(3*n^2+3*n)*E-3*n^3-3*n^2;
> efactor(L, E, n);

$$\left[1,{\it LeftLcm}(E+-i\sqrt {3}n,E+i\sqrt {3}n),E-n\right]$$

See Also

decompose,Loewy