Some remarks on the Jacobian conjecture and polynomial endomorphisms

Abstract:

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization which states 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 $\ker \mathcal {J} H$ has $n-r$ independent vectors over the base field, in particular for invertible power linear maps $x + (Ax)^{*d}$ with $\operatorname {rk} A = r$, the degree of the inverse of $x + H$ is at most $d^r$.