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The Analogue of Büchi’s Problem for Polynomials

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New Computational Paradigms (CiE 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3526))

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Abstract

Büchi’s problem asked whether a surface of a specific type, defined over the rationals, has integer points other than some known ones. A consequence of a positive answer would be the following strengthening of the negative answer to Hilbert’s tenth problem : the positive existential theory of the rational integers in the language of addition and a predicate for the property ‘x is a square’ would be undecidable. Despite some progress, including a conditional positive answer (pending on conjectures of Lang), Büchi’s problem remains open.

In this article we prove an analogue of Büchi’s problem in rings of polynomials of characteristic either 0 or p ≥ 13.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Crete, Knossos Avenue, 714 09, Iraklio, Greece

    Thanases Pheidas

  2. Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK

    Xavier Vidaux

Authors
  1. Thanases Pheidas
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  2. Xavier Vidaux
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Editor information

Editors and Affiliations

  1. School of Mathematics, University of Leeds, LS2 9JT, Leeds, U.K.

    S. Barry Cooper

  2. Department Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Benedikt Löwe

  3. Universiteit van Amsterdam,  

    Leen Torenvliet

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Pheidas, T., Vidaux, X. (2005). The Analogue of Büchi’s Problem for Polynomials. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_50

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  • DOI: https://doi.org/10.1007/11494645_50

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26179-7

  • Online ISBN: 978-3-540-32266-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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