A note on the kernels of higher derivations

Abstract

Let k ⊆ k′ be a field extension. We give relations between the kernels of higher derivations on k[X] and k′[X], where k[X]:= k[x ₁,…, x _n ] denotes the polynomial ring in n variables over the field k. More precisely, let D = {D _n } ^∞ _n=0 a higher k-derivation on k[X] and D′ = {D′ _n } ^∞ _n=0 a higher k′-derivation on k′[X] such that D′ _m (x _i ) = D _m (x _i ) for all m ⩾ 0 and i = 1, 2,…, n. Then (1) k[X]^D = k if and only if k′[X]^D′ = k′; (2) k[X]^D is a finitely generated k-algebra if and only if k′[X]^D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[X]^D of a higher derivation D of k[X] can be generated by a set of closed polynomials.