symmetricKernel

Usage

symmetricKernel(L, n)

Parameter	Type	Description
L	$\mathbbm{Q}[x,\frac d{dx}]$	A differential operator
n	$\mathbbm{Z}$	A positive integer

Returns

symmetricKernel L returns a basis for $\mbox{Ker}{\left({\mbox{ ${L}^{\raise2pt\vbox{\hbox{$\mbox{\footnotesize\vbox{\... ...}}}$}} \raise3pt\vbox{\hbox{$\scriptstyle n$}}} $}}\right)} \cap \mathbbm{Q}(x)$, where $\mbox{ ${L}^{\raise2pt\vbox{\hbox{$\mbox{\footnotesize\vbox{\hbox{$\mathop{\mat... ...box{\hbox{$s$}}\mskip3mu}}\,$}}}$}} \raise3pt\vbox{\hbox{$\scriptstyle n$}}} $}$ is the $n^{{\rm th}}$ symmetric power of $L$.

Remarks

Using symmetricKernel can be more efficient than computing the symmetric power and its kernel separately.

Example

We compute the rational kernel of the second symmetric power of

$$L = \frac{d^7}{dx^7} - x \frac d{dx} + \frac 12$$

as follows:

1 --> L := D^7 - x*D + 1/2;
2 --> K := symmetricKernel(L, 2);
3 --> tex(K);

$$[ x ]$$

See Also

kernel, symmetricPower