A Groebner basis approach to solve a Conjecture of Nowicki

Let k be a field of characteristic zero, n any positive integer and let n be the derivation n i=1 Xi 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 n = k[X1, . . . , Xn, XiYj - XjYi; 1 i < j n] in which case we say that n 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=1 Xti i Yi , ti Z0. 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. Key words: Locally nilpotent derivations, Elimination theory.

• Type: Journal
• Year: 2008
• Where: JSC
• Authors: Joseph Khoury
• Keywords: Characteristic Zero, JSC 2008, Positive Integer, Xi Yi