We present a generalization of the Jacobian Conjecture for m polynomials in n variables:, where k is a field of characteristic zero and m ∈ {1, . . . , n}. We express the generalized Jacobian condition in terms of irreducible and square-free elements of the subalgebra k[f 1 , . . . , f m ]. We also discuss obtained properties in a more general setting -for subrings of unique factorization domains.

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We obtain two equivalent conditions for polynomials in variables to form a -basis of a ring of constants of some polynomial K -derivation, where K is a unique factorization domain of characteristic > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case = this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic. MSC:13N15, 13F20, 14R15