Rings and fields of constants of cyclic factorizable derivations

Abstract

We present a survey of the research on rings of polynomial constants and fields of rational constants of cyclic factorizable derivations in polynomial rings over fields of characteristic zero. 1. Motivations and preliminaries The first inspiration for the presented series of articles (some of them are joint works with Hegedűs and Ossowski) was the publication [20] of professor Nowicki and professor Moulin Ollagnier. The fundamental problem investigated in that series of articles concerns rings of polynomial constants ([26], [28], [33], [29], [8]) and fields of rational constants ([30], [31], [32]) in various classes of cyclic factorizable derivations. Moreover, we investigate Darboux polynomials of such derivations together with their cofactors ([33]) and applications of the results obtained for cyclic factorizable derivations to monomial derivations ([31]). Let k be a field. If R is a commutative k-algebra, then k-linear mapping d : R→ R is called a k-derivation (or simply a derivation) of R if d(ab) = ad(b) + bd(a) for all a, b ∈ R. The set R = ker d is called a ring (or an algebra) of constants of the derivation d. Then k ⊆ R and a nontrivial constant of the derivation d is an element of the set R \ k. By k[X] we denote k[x1, . . . , xn], the polynomial ring in n variables. If f1, . . . , fn ∈ k[X], then there exists exactly one derivation d : k[X]→ k[X] such that d(x1) = f1, . . . , d(xn) = fn. 2010 Mathematics Subject Classification. 13N15, 12H05, 34A34.