Constants of cyclotomic derivations

Abstract

Let k[X]=k[x0,…,xn−1] and k[Y]=k[y0,…,yn−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 Δ be the factorisable derivation of k[Y] associated with d, that is, d(xj)=xj+1 and Δ(yj)=yj(yj+1−yj) for all j∈Zn. 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−φ(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 Δ is always equal to k[v], where v is the product y0⋯yn−1, and we describe the field of constants of Δ in two cases: when n is power of a prime, and when n=pq.