Analogs of Jacobian conditions for subrings

Abstract

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.

On properties of square-free elements in commutative cancellative monoids

“…Thus we have the equivalence of the conditions (vi) -(viii) of the following Theorem 5.1. For the same reason implication (viii) ⇒ (i) of Theorem 5.1 follows from the proof of implication (ii) ⇒ (i) of Theorem 3.4 from [15]. (vi) for every n ≥ 1, k 1 , .…”

“…. , f r are algebraically independent over k, the generalized Jacobian condition (2.5) is equivalent to any of the following ones ( [15]):…”

“…If H is a factorial monoid and a submonoid M ⊂ H satisfies M × = H × and q(M) ∩ H = M, then condition (1.2) can be expressed in a factorial way (see [15], Theorem 3.4 -formulated in terms of rings, but in fact valid for monoids): Recall also (see [15], Theorem 3.6) that under these assumptions a submonoid M satisfying (1.2) is root closed in H. Recently Angermüller showed in [4], Proposition 32, that under the same assumptions a submonoid M satisfying (1.1) is root closed in H. A submonoid M ⊂ H is called root closed in H if, for every a ∈ H and n ≥ 1, a n ∈ M implies a ∈ M.…”

An approach to the Jacobian Conjecture in terms of irreducibility and square-freeness

“…A generalization of Freudenburg's Lemma to an arbitrary number of polynomials over a field of characteristic zero was obtained in [24]. Denote by jac f 1 ,...,fr…”

“…for subrings of unique factorization domains. In this case the property that square-free elements of a subring are square-free in the whole ring can be expressed in some factorial form ( [24], Theorem 3.4). At the end we discuss possible directions of future research.…”