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The Weitzenboeck theorem states that the algebra of constants of a linearlocally nilpotent derivation of the polynomial algebra K[Z]=K[z_1,...,z_m] in mvariables over a field K of characteristic 0 is finitely generated. If m=2n andthe Jordan normal form of the derivation consists of Jordan cells of size 2only, we may assume that K[Z]=K[X,Y] and the derivation sends y_i to x_i andx_i to 0, i=1,...,n. Nowicki conjectured that the algebra of constants of thisderivation is generated by x_1,...,x_n and x_iy_j-x_jy_i, i<j. Recently thisconjecture was confirmed in the Ph.D. thesis of Khoury, and also by Derksen. Inthis paper we give an elementary proof of the conjecture of Nowicki. Then wefind a very simple system of defining relations of the algebra of constantswhich corresponds to the reduced Groebner basis of the related ideal withrespect to a suitable admissible order, and present an explicit basis of thealgebra of constants as a vector space.

THE CONJECTURE OF NOWICKI ON WEITZENBÖCK DERIVATIONS OF POLYNOMIAL ALGEBRAS