A characterization of Keller maps

TL;DR

This paper explores the Jacobian Conjecture by characterizing when polynomial endomorphisms map irreducible polynomials to square-free polynomials, linking this property to automorphisms in polynomial algebra.

Contribution

It establishes that the Jacobian condition is equivalent to mapping irreducible polynomials to square-free polynomials, offering a new perspective on the Jacobian Conjecture.

Findings

Jacobian condition equivalent to square-free polynomial mapping

Characterization of automorphisms via polynomial irreducibility

Reformulation of the Jacobian Conjecture in terms of square-free polynomials

Abstract

Let k be a field of characteristic zero. Let phi be a k-endomorphism of the polynomial algebra k[x_1,...,x_n]. It is known that phi is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. In this paper we show that phi satisfies the jacobian condition if and only if it maps irreducible polynomials to square-free polynomials. Therefore, the Jacobian Conjecture is equivalent to the following statement: every k-endomorphism of k[x_1,...,x_n], mapping irreducible polynomials to square-free polynomials, maps irreducible polynomials to irreducible polynomials.