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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An extension of Büchi's problem for polynomial rings in zero characteristic


Author: Hector Pasten
Journal: Proc. Amer. Math. Soc. 138 (2010), 1549-1557
MSC (2010): Primary 11U05, 12L05; Secondary 11C08
DOI: https://doi.org/10.1090/S0002-9939-09-10259-9
Published electronically: December 29, 2009
MathSciNet review: 2587438
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Abstract: We prove a strong form of the ``$ n$ Squares Problem'' over polynomial rings with characteristic zero constant field. In particular we prove : for all $ r\ge 2$ there exists an integer $ M=M(r)$ depending only on $ r$ such that, if $ z_1,z_2,...,z_M$ are $ M$ distinct elements of $ F$ and we have polynomials $ f,g,x_1,x_2,\dots,x_M\in F[t]$, with some $ x_i$ non-constant, satisfiying the equations $ x_i^r=(z_i+f)^r+g$ for each $ i$, then $ g$ is the zero polynomial.


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Additional Information

Hector Pasten
Affiliation: Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile
Email: hpasten@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-09-10259-9
Keywords: B\"uchi's problem, squares problem, polynomials, Hilbert's tenth problem
Received by editor(s): September 2, 2008
Received by editor(s) in revised form: March 12, 2009
Published electronically: December 29, 2009
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.