The Conjecture of Nowicki on Weitzenboeck derivations of polynomial algebras

TL;DR

This paper provides an elementary proof of Nowicki's conjecture on the generators of the algebra of constants for a specific class of Weitzenboeck derivations, and describes its defining relations and basis explicitly.

Contribution

It offers a new elementary proof of Nowicki's conjecture and explicitly characterizes the algebra of constants with simple relations and a basis.

Findings

Confirmed Nowicki's conjecture for the algebra of constants.

Derived a simple system of defining relations for the algebra.

Presented an explicit basis of the algebra as a vector space.

Abstract

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.