Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : kX] → kX] and d ₂ : kY] → kY] be k-derivations, where kX] ? kx ₁,…,x _n ], kY] ? ky ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of kX, Y] such that d| _kX] = d ₁ and d| _kY] = 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 kY] 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.