The Conjecture of Nowicki on Weitzenboeck derivations of polynomial algebras

The Weitzenboeck theorem states that the algebra of constants of a linear locally nilpotent derivation of the polynomial algebra K[Z]=K[z_1,...,z_m] 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 y_i to x_i and x_i to 0, i=1,...,n. Nowicki conjectured that the algebra of constants of this derivation is generated by x_1,...,x_n and x_iy_j-x_jy_i, 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.