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.
We present some general properties of the field of constants of monomial derivations of k(x 1 , . . . , x n ), where k is a field of characteristic zero. The main result of this paper is a description of all monomial derivations of k(x, y, z) with trivial field of constants. In this description a crucial role plays the classification result of Moulin Ollagnier for Lotka-Volterra derivations with strict Darboux polynomials. Several applications of our description are also given in this paper.
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