A Groebner basis approach to solve a Conjecture of Nowicki

Let kk be a field of characteristic zero, nn any positive integer and let δnδn be the derivation ∑i=1nXi∂∂Yi of the polynomial ring k[X1,…,Xn,Y1,…,Yn]k[X1,…,Xn,Y1,…,Yn] in 2n2n variables over kk. A Conjecture of Nowicki (Conjecture 6.9.10 in [Nowicki, A. 1994. Polynomial derivations and their rings of constants, Wydawnictwo Uniwersytetu Mikolaja Kopernika, Torun]) states the following kerδn=k[X1,…,Xn,XiYj−XjYi;1≤i