Irreducibility properties of Keller maps

TL;DR

This paper explores the irreducibility properties of Keller maps over fields of characteristic zero, connecting these properties to the Jacobian Conjecture and extending previous results on polynomial map invertibility.

Contribution

It generalizes the classification of Keller maps by irreducibility properties and shows the Jacobian Conjecture can be reduced to specific types with irreducibility conditions.

Findings

Endomorphisms map irreducible polynomials to square-free polynomials.

Components of cubic homogeneous Keller maps with symmetric Jacobian matrices are irreducible.

Jacobian Conjecture reduces to Keller maps with irreducible affine combinations.

Abstract

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducible properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski, that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.