The Conjecture of Nowicki on Weitzenböck Derivations of Polynomial Algebras

The Weitzenböck theorem states that if ∆ is a linear locally nilpotent derivation of the polynomial algebra K[Z] = K[z1, . . . , zm] over a field K of characteristic 0, then the algebra of constants of ∆ is finitely generated. If m = 2n and the Jordan normal form of ∆ consists of 2 × 2 Jordan cells only, we may assume that K[Z] = K[X,Y ] and ∆(yi) = xi, ∆(xi) = 0, i = 1, . . . , n. Nowicki conjectured that the algebra of constants K[X, Y ] is generated by x1, . . . , xn and xiyj − xjyi, 1 ≤ i < j ≤ n. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury, and also by Derksen. In this paper we give an elementary proof of the conjecture of Nowicki. Then we find a very simple system of defining relations of the algebra K[X,Y ] which corresponds to the reduced Gröbner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of K[X, Y ] as a vector space. Introduction Let K be a field of characteristic 0 and let K[Z] = K[z1, . . . , zm] be the polynomial K-algebra in m variables. A linear operator ∆ of K[Z] is called a derivation if ∆(uv) = ∆(u)v + u∆(v) for all u, v ∈ K[Z]. The derivation ∆ is locally nilpotent if for each u ∈ K[Z] there exists a d ≥ 1 such that ∆(u) = 0. Locally nilpotent derivations of polynomial algebras are subjects of active investigation. They play essential role in the study of automorphism group of K[Z], including the generation of Aut K[x, y] by tame automorphisms, the Jacobian conjecture, in invariant theory, Fourteenth Hilbert’s problem and other important topics. See the books by Nowicki [N], van den Essen [E], and Freudenburg [F] for details. The well known theorem of Weitzenböck [W] states that if ∆ is a nilpotent linear operator acting on the m-dimensional vector space KZ = Kz1 ⊕ · · · ⊕ Kzm and we extend it to a derivation of K[Z], then the algebra K[Z] of constants of ∆, i.e., the kernel of ∆ in K[Z], is a finitely generated algebra. We call ∆, which is a locally nilpotent derivation, a Weitzenböck derivation. A modern proof of the theorem of Weitzenböck is given by Seshadri [S], with further simplification by Tyc [T], see also [N, F]. Up to a change of the basis of the vector space KZ, Weitzenböck derivations are determined by their Jordan normal form. Each Jordan cell is an upper triangular matrix with zero diagonal. Hence, for each dimension m it is sufficient to consider a finite number of Weitzenböck derivations ∆. There are algorithms which find a set of generators of K[Z] for a given ∆. Nevertheless from computational point 2000 Mathematics Subject Classification. 13N15; 13A50; 13P10; 14E07. The research of the first author was partially supported by Grant MI-1503/2005 of the Bulgarian National Science Fund. The work of the second author was partially supported by an NSA grant.