Derivations of polynomial algebras without Darboux polynomials

We present several new examples of homogeneous derivations of a polynomial ring k[X] = k[x(1),...,x(n)] over a field k of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number n of variables, thus more easily than the previously known example multidimensional Jouanolou systems of Zoladek. (C) 2007 Elsevier B.V. All rights reserved.