Constants of cyclotomic derivations

Abstract

Let $k[X]=k[x_0,...,x_{n-1}]$ and $k[Y]=k[y_0,...,y_{n-1}]$ be the polynomial rings in $n⩾3$ variables over a field $k$ of characteristic zero containing the $n$-th roots of unity. Let $d$ be the cyclotomic derivation of $k[X]$, and let $\Delta$ be the factorisable derivation of $k[Y]$ associated with $d$, that is, $d(x_j)=x_{j+1}$ and $\Delta (y_j)=y_j(y_{j+1}-y_j)$ for all $j\in {\mathbb{Z}}_n$. We describe polynomial constants and rational constants of these derivations. We prove, among others, that the field of constants of $d$ is a field of rational functions over $k$ in $n-\f(n)$ variables, and that the ring of constants of $d$ is a polynomial ring if and only if $n$ is a power of a prime. Moreover, we show that the ring of constants of $\Delta$ is always equal to $k[v]$, where $v$ is the product $y_0...y_{n-1}$, and we describe the field of constants of $\Delta$ in two cases: when $n$ is power of a prime, and when $n=pq$.