Analogs of Jacobian conditions for subrings

TL;DR

This paper generalizes the Jacobian Conjecture to multiple polynomials in several variables, expressing conditions via irreducible and square-free elements in subalgebras, and extends properties to subrings of UFDs.

Contribution

It introduces a new generalized Jacobian condition for subrings generated by multiple polynomials and explores its properties in the context of UFDs.

Findings

Generalized Jacobian condition expressed through irreducible elements

Extension of properties to subrings of UFDs

Potential new criteria for polynomial invertibility

Abstract

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], 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[f1,...,fm]. We also discuss obtained properties in a more general setting - for subrings of unique factorization domains.