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 ∑n i=1Xi ∂ ∂Yi of the polynomial ring k[X1, . . . , Xn, Y1, . . . , Yn] in 2n variables over k. A Conjecture of Nowicki (Conjecture 6.9.10 in (8)) states the following ker δn = k[X1, . . . , Xn, XiYj −XjYi; 1 ≤ i < j ≤ n] 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 D = ∑n i=1X ti i ∂ ∂Yi , ti ∈ Z≥0. In (6), H. Kojima and M. 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 Section 4 of this paper that the proof presented in (6) is not complete.