Rings of constants of polynomial derivations and p-bases

Abstract

Abstract. We present a survey of results concerning p-bases of rings of constants with respect to polynomial derivations in characteristic p > 0. We discuss characterizations of rings of constants, properties of their generators and a general characterization of their p-bases. We also focus on some special cases: one-element p-bases, eigenvector p-bases and when a ring of constants is a polynomial graded subalgebra. IntroductionIn Section 1 we introduce the notation and definitions concerning derivations, rings of constants and p-bases. Then we discuss characterizations of rings of constants in Section 2 and we present some basic information on the number of generators for rings of constants of polynomial derivations in Section 3. For a wider panorama of contemporary differential algebra we refer to the book of Nowicki ([41]), and for problems connected with locally nilpotent derivations we refer to the book of Freudenburg ([10]).Next two sections contain a general characterization of p-bases of rings of constants with respect to polynomial derivations, based on the author's paper [26]. In Section 4 we present generalizations of Freudenburg's lemma (Theorems 4.7 and 4.8). The main theorem (Theorem 5.4) and its motivations are presented in Section 5. In Section 6 (based on the results of [23] and [18]) we discuss analogies and differences between single generators of rings of constants in zero and positive characteristic, and we focus on some special cases. Section 7, based on [24], is devoted to specific properties of eigenvector p-bases (Theorem 7.2). Finally, in 2010 Mathematics Subject Classification. Primary 13N15, Secondary 13F20.