Picard-vessiot theory and the Jacobian problem

Abstract

A differential version of the classical theorem of Campbell is presented In this paper we establish the equivalence of the two existing definitions of Picard–Vessiot extension of partial differential fields and give an equivalent statement of the Jacobian conjecture in terms of Picard–Vessiot extensions. We denote as usual by K〈S〉 the differential field differentially generated by the set S, and by CK the field of constants of a differential field K. Kolchin gave in [K] a definition of Picard–Vessiot extension for partial differential fields which is a generalization of the definition of Picard–Vessiot extension associated with a linear differential operator defined over an ordinary differential field. With a partial differential field extension L|K, with derivations ∂1, . . . , ∂m, he associates an ordinary differential field extension LD|KD, where KD := K〈u1, . . . , um〉, LD := L〈u1, . . . , um〉, with u1, . . . , um independent differential indeterminates, and the derivation is D := u1∂1 + · · ·+ um∂m. Received November 18, 2009