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Proceedings of the American Mathematical Society

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Some remarks on the Jacobian conjecture and polynomial endomorphisms

Authors: Dan Yan and Michiel de Bondt
Journal: Proc. Amer. Math. Soc. 142 (2014), 391-400
MSC (2010): Primary 14E05; Secondary 14A05, 14R15
Published electronically: October 30, 2013
MathSciNet review: 3133981
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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$.

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Additional Information

Dan Yan
Affiliation: School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

Michiel de Bondt
Affiliation: Department of Mathematics, Radboud University, Nijmegen, The Netherlands

Keywords: Jacobian conjecture, polynomial map, Dru{\.z}kowski map
Received by editor(s): March 9, 2012
Received by editor(s) in revised form: March 15, 2012, and March 23, 2012
Published electronically: October 30, 2013
Additional Notes: The second author was supported by the Netherlands Organisation for Scientific Research (NWO)
Communicated by: Lev Borisov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.