symmetricPower

Usage

symmetricPower(L, n)

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

Returns

Returns a differential operator $\mbox{ ${L}^{\raise2pt\vbox{\hbox{$\mbox{\footnotesize\vbox{\hbox{$\mathop{\mat... ...box{\hbox{$s$}}\mskip3mu}}\,$}}}$}} \raise3pt\vbox{\hbox{$\scriptstyle n$}}} $}$ of minimal order whose kernel is generated by the all the products of $n$ elements of a basis of $\mbox{Ker}(L)$.

Remarks

If you only want to compute the kernel of $\mbox{ ${L}^{\raise2pt\vbox{\hbox{$\mbox{\footnotesize\vbox{\hbox{$\mathop{\mat... ...box{\hbox{$s$}}\mskip3mu}}\,$}}}$}} \raise3pt\vbox{\hbox{$\scriptstyle n$}}} $}$ but not necessarily the symmetric power itself, then use symmetricKernel instead.

Example

The kernel of the fourth symmetric power of

$$L = x^2 \frac{d^2}{dx^2} + \frac 3{16} - x$$

can be computed as follows:

1 --> L := x^2*D^2 + 3/16 - x;
2 --> L4 := symmetricPower(L, 4);
3 --> tex(L4);

$$\begin{eqnarray*} 4\,x^{5}\,D^{5}&+&\left(-80\,x^{4}+15\,x^{3}\right)\,D^{3}+ \l... ...}-240\,x^{2}+90\,x\right)\,D+ \left(-256\,x^{2}+240\right)\,x-90 \end{eqnarray*}$$

4 --> K := kernel(L4);
5 --> tex(K);

$$[ x ]$$

See Also

exteriorPower, symmetricKernel