Birational morphisms of the plane

Authors:
Vladimir Shpilrain and Jie-Tai Yu

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2511-2515

MSC (2000):
Primary 14E07, 14E25; Secondary 14A10, 13B25

DOI:
https://doi.org/10.1090/S0002-9939-04-07490-8

Published electronically:
April 8, 2004

MathSciNet review:
2054774

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the affine plane over a field of characteristic . Birational morphisms of are mappings given by polynomial mappings of the polynomial algebra such that for the quotient fields, one has . Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping given by . For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of . This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.

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Additional Information

**Vladimir Shpilrain**

Affiliation:
Department of Mathematics, The City College of New York, New York, New York 10031

Email:
shpil@groups.sci.ccny.cuny.edu

**Jie-Tai Yu**

Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Email:
yujt@hkusua.hku.hk

DOI:
https://doi.org/10.1090/S0002-9939-04-07490-8

Keywords:
Affine plane,
birational morphisms,
peak reduction

Received by editor(s):
November 13, 2002

Published electronically:
April 8, 2004

Additional Notes:
The second author was partially supported by RGC Grant Project 7126/98P

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2004
American Mathematical Society