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 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 $x+(Ax{)}^{\ast d}$ with $\rk A = r$, the degree of the inverse of $x+H$ is at most $d^r$.

11 pages

References in corpus (1)

• The Amazing Image Conjecture

Cited by in corpus (5)

• Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings
• Some approaches toward the Jacobian conjecture
• Jacobian conjecture in ${\mathbb{R}}^2$
• Some observations on the properness of identity plus linear powers: part 2
• A particular case of the Jacobian conjecture