The Conjecture of Nowicki on Weitzenboeck derivations of polynomial algebras
Abstract
The Weitzenboeck theorem states that the algebra of constants of a linear locally nilpotent derivation of the polynomial algebra K[Z]=K[z1,...,zm] in m variables over a field K of characteristic 0 is finitely generated. If m=2n and the Jordan normal form of the derivation consists of Jordan cells of size 2 only, we may assume that K[Z]=K[X,Y] and the derivation sends yi to xi and xi to 0, i=1,...,n. Nowicki conjectured that the algebra of constants of this derivation is generated by x1,...,xn and xiyj-xjyi, i<j. 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 of constants which corresponds to the reduced Groebner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of the algebra of constants as a vector space.