# Symmetric Powers

## `SymmetricPower(L, m): RngDiffOpElt, RngIntElt -> RngDiffOpElt`

Returns the $m$-th symmetric power of the differential operator $L$ as an element of the parent of $L$. The symmetric power is monic where possible. If $n$ denotes the order of $L$, then the degree of the $m$-th symmetric power of $L$ is at most ${n+m-1}\choose{n-1}$. The algorithm that is used is based on algorithms given in [[Bronstein *et al.*, 1997](../../references.md#cite-bmw97)].

## `Example Symmetric Power (ex-6160ae)`

```magma
> F<z> := RationalDifferentialField(Rationals());
> R<D> := DifferentialOperatorRing(F);
> SymmetricPower(D^2, 3);
D^4
> SymmetricPower(D^3-1, 2);
D^6 + -7*D^3 + -8

```