A Groebner basis approach to solve a Conjecture of Nowicki

Let k be a field of characteristic zero, n any positive integer and let δn be the derivation of the polynomial ring k[X1,…,Xn,Y1,…,Yn] in 2n variables over k. A Conjecture of Nowicki (Conjecture 6.9.10 in [Nowicki, A. 1994. Polynomial derivations and their rings of constants, WydawnictwoUniwersytetuMikolajaKopernika, Torun]) states the following in which case we say that δn is standard. In this paper, we use the elimination theory of Groebner bases to prove that Nowicki’s conjecture holds in the more general case of the derivation , . In [Kojima, H. Miyanishi, M. 1997. On Robert’s counterexample to the fourteenth problem of Hilbert, J. Pure Appl. Algebra 122, 277–292], Kojima and Miyanishi argued that D is standard in the case where ti=t (i=1,…n) for some t≥3. Although the result is true, we show in this paper that their proof is not complete.