2008/04/18 by Vesselin Drensky, Drensky, Vesselin, Leonid Makar-Limanov +1
Computer Science · Mathematics · #13A50 #13N15 #13P10 #14E07 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #math.AC #msc:13A50 #msc:13N15 #msc:13P10 #msc:14E07
paper · pdf · doi:10.48550/arxiv.0804.2933
8 pages
arxiv created 2008/04/18 · openalex publication_date 2008/04/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.