Abstract: | Abstract Let d 1 : k[X] → k[X] and d 2 : k[Y] → k[Y] be k-derivations, where k[X] ? k[x 1,…,x n ], k[Y] ? k[y 1,…,y m ] are polynomial algebras over a field k of characteristic zero. Denote by d 1 ⊕ d 2 the unique k-derivation of k[X, Y] such that d| k[X] = d 1 and d| k[Y] = d 2. We prove that if d 1 and d 2 are positively homogeneous and if d 1 has no nontrivial Darboux polynomials, then every Darboux polynomial of d 1 ⊕ d 2 belongs to k[Y] and is a Darboux polynomial of d 2. We prove a similar fact for the algebra of constants of d 1 ⊕ d 2 and present several applications of our results. |