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POLYNOMIAL ENDOMORPHISMS PRESERVING OUTER RANK IN TWO VARIABLES

Part of: Affine geometry

Published online by Cambridge University Press:  16 February 2012

YONG JIN  and
XIANKUN DU
YONG JIN
Affiliation:
School of Mathematics, Jilin University, 130012 Changchun, PR China (email: kingmeng@126.com)
XIANKUN DU*
Affiliation:
School of Mathematics, Jilin University, 130012 Changchun, PR China (email: duxk@jlu.edu.cn)
*
✉For correspondence; e-mail: kingmeng@126.com
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Abstract

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An endomorphism φ of a polynomial ring is said to preserve outer rank if φ sends each polynomial to one with the same outer rank. For the polynomial ring in two variables over a field of characteristic 0 we prove that an endomorphism φ preserving outer rank is an automorphism if one of the following conditions holds: (1) the Jacobian of φ is a nonzero constant; (2) the image of φ contains a coordinate; (3) φ has a ‘fixed point’.

Keywords

outer rankcoordinatetest polynomialretract

MSC classification

Secondary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 186 - 192
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

Supported by NSF of China (No.11071097) and ‘211 Project’ and ‘985 Project’ of Jilin University.

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