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CLASSIFICATION OF CUBIC HOMOGENEOUS POLYNOMIAL MAPS WITH JACOBIAN MATRICES OF RANK TWO

Part of: Affine geometry

Published online by Cambridge University Press:  30 May 2018

MICHIEL DE BONDT  and
XIAOSONG SUN Open the ORCID record for XIAOSONG SUN [Opens in a new window]
MICHIEL DE BONDT
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, The Netherlands email [email protected]
XIAOSONG SUN*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, China email [email protected]
*
✉[email protected]
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Abstract

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Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.

Keywords

cubic homogeneous polynomial mapJacobian conjectureKeller maprank-two maptame map

MSC classification

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Secondary: 14R15: Jacobian problem
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 89 - 101
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author has been supported by the Netherlands Organisation of Scientific Research (NWO). The second author has been partially supported by the NSF of China (grant nos. 11771176 and 11601146) and by the China Scholarship Council.

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