Constants of derivations in polynomial rings over unique factorization domains

Abstract

Abstract. A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any K-derivation of K [x, y], where K is a commutative field of characteristic zero, is a polynomial ring in one variable over K. In this paper we give an elementary proof of this theorem and show that it remains true if we replace K by any unique factorization domain of characteristic zero.

On the kernels of higher R-derivations of R[x1,…,xn]

“…The above result on the kernels of higher derivations is same as Theorem 2.8 of Nowicki and Nagata [6] and Corollary 3.2 of Kahoui [3] in the context of ordinary derivation. There are many other interesting results on derivations which have been extended to higher derivations.…”

“…. The above result is generalised by Berson in [2] and by Kahoui in [3] to the polynomial ring R[x 1 , x 2 ] over a unique factorization domain R. More precisely, they proved that if R is a unique factorization domain and d is a non trivial R-derivation of…”

Triangulable locally nilpotent derivations in dimension three

“…As a consequence of theorem 2.2, if A is a UFD containing Q and X is a locally nilpotent A-derivation of A[x, y] then there exists f ∈ A[x, y] and a univariate polynomial [9,12].…”

Closed polynomials and their applications for computations of kernels of monomial derivations

“…In the case where R is a UFD, the kernel of a non-zero derivation on R[x, y] is generated by one polynomial (see [3,Corollary 3.2]) and it is integrally closed in R[x, y]. Thus, if R[x, y] D = R, then it is generated by a closed polynomial.…”

“…It is well-known that the kernel of any k-derivation on k[X] with n ≤ 3 is finitely generated as a k-algebra and that the kernel of any non-zero k-derivation on k[X] with n = 2 can be expressed as k[f ] for some f ∈ k[X] D , which are originally given in [14]. Note also, the fact holds true in the case where k is a UFD of characteristic zero and n = 2 (see [3,Corollary 3.2]). However, it is difficult to determine the generator of k[X] D of some k-derivation D on k[X] even if k[X] D is finitely generated as a k-algebra.…”