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Representation of squares by monic second degree polynomials in the field of $ p$-adic meromorphic functions

Author: Hector Pasten
Journal: Trans. Amer. Math. Soc. 364 (2012), 417-446
MSC (2010): Primary 11U05, 30D35; Secondary 30D30
Published electronically: August 23, 2011
MathSciNet review: 2833587
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Abstract: We prove a result on the representation of squares by monic second degree polynomials in the field of $ p$-adic meromorphic functions in order to solve positively Büchi's $ n$ squares problem in this field. Using this result, we prove the non-existence of an algorithm to decide whether a system of diagonal quadratic forms over $ \mathbb{Z}[z]$ represents or not in the ring of $ p$-adic entire functions (in the variable $ z$) a given vector of polynomials in $ \mathbb{Z}[z]$, and a similar result for $ p$-adic meromorphic functions when the systems allow vanishing conditions on the unknowns. This improves the known negative answers for the analogue of Hilbert's Tenth Problem for these structures. We also improve some results by Vojta concerning the case of complex meromorphic functions, the case of function fields and finally the case of number fields, and we show an intimate relation of the latter with Bombieri's Conjecture for surfaces over number fields.

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Additional Information

Hector Pasten
Affiliation: Department of Mathematics, Universidad de Concepción, Chile
Address at time of publication: Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

Keywords: $p$-adic meromorphic, Büchi’s problem, Hilbert’s Tenth Problem
Received by editor(s): July 12, 2010
Published electronically: August 23, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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