A characterization of Keller maps

Abstract

Let k be a field of characteristic zero. Let φ be a k-endomorphism of the polynomial algebra k[x1,…,xn]. It is known that φ is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. In this paper we show that φ 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[x1,…,xn], mapping irreducible polynomials to square-free polynomials, maps irreducible polynomials to irreducible polynomials.