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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: April 8, 2004
MathSciNet review: 2054774
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Abstract: Let $A^2$ be the affine plane over a field $K$ of characteristic $0$. Birational morphisms of $A^2$ are mappings $A^2 \to A^2$ given by polynomial mappings $\varphi$ of the polynomial algebra $K[x,y]$ such that for the quotient fields, one has $K(\varphi(x), \varphi(y)) = K(x,y)$. Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping $\tau_x$ given by $x \to x, ~y \to xy$. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of $\tau_x$. 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: http://dx.doi.org/10.1090/S0002-9939-04-07490-8
PII: S 0002-9939(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