Some remarks on the Jacobian conjecture and polynomial endomorphisms

TL;DR

This paper investigates properties of polynomial maps related to the Jacobian conjecture, establishing injectivity conditions, value restrictions on collinear points, and degree bounds for inverses of certain invertible maps.

Contribution

It introduces new injectivity results for homogeneous Keller maps, generalizes conditions preventing repeated values on lines, and provides degree bounds for inverses of specific invertible polynomial maps.

Findings

Homogeneous Keller maps are injective on lines through the origin.

Polynomial endomorphisms with multiple homogeneous parts do not share the same image on collinear points under certain conditions.

The degree of the inverse of specific invertible maps is bounded by a power of the original degree.

Abstract

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of degree $d$, such that $\ker \jac H$ has $n-r$ independent vectors over the base field, in particular for invertible power linear maps $x + (Ax)^{*d}$ with $\rk A = r$, the degree of the inverse of $x + H$ is at most $d^r$.