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Uniform existential interpretation of arithmetic in rings of functions of positive characteristic

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Abstract

We show that first order integer arithmetic is uniformly positive-existentially interpretable in large classes of (subrings of) function fields of positive characteristic over some languages that contain the language of rings. One of the main intermediate results is a positive existential definition (in these classes), uniform among all characteristics p, of the binary relation “\(y=x^{p^{s}}\) or \(x=y^{p^{s}}\) for some integer s≥0”. A natural consequence of our work is that there is no algorithm to decide whether or not a system of polynomial equations over \(\mathbb {Z}[z]\) has solutions in all but finitely many polynomial rings \(\mathbb {F}_{p}[z]\). Analogous consequences are deduced for the rational function fields \(\mathbb {F}_{p}(z)\), over languages with a predicate for the valuation ring at zero.

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Notes

  1. We have been informed that they now have obtained (non-uniform) positive existential such definitions over \(\mathcal {L}_{z,{\rm ord}}\).

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Acknowledgements

The authors would like to thank Ricardo Baeza, Antonio Laface, Angus Macintyre and Thomas Scanlon for comments on a first version of this paper, and Alexander Molnar for a careful reading of the very final version of it. The authors were also benefited from discussions with Ram Murty, Alexandra Shlapentokh and Carlos Videla.

Finally, the authors would like to heartily thank the anonymous referee. Her or his comments and corrections greatly improved the quality and presentation of this work.

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Authors and Affiliations

  1. Queen’s University, Kingston, ON, Canada, K7L 3N6

    Hector Pasten

  2. University of Crete and Institute of Applied and Computational Mathematics, Foundation of Research and Technology, 71409, Heraklion, Crete, Greece

    Thanases Pheidas

  3. Universidad de Concepción, Concepción, Chile

    Xavier Vidaux

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  1. Hector Pasten
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  2. Thanases Pheidas
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  3. Xavier Vidaux
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Correspondence to Thanases Pheidas.

Additional information

This work was developed as part of the first author’s thesis at Universidad de Concepción, (Chile). It was supported by the third author’s Chilean research project Fondecyt 1090233. During the revision of this work, the first author was partially supported by an Ontario Graduate Scholarship, and the second author was supported by the Conicyt Program ‘Atracción de Capital Humano Avanzado’ 80112001.

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Pasten, H., Pheidas, T. & Vidaux, X. Uniform existential interpretation of arithmetic in rings of functions of positive characteristic. Invent. math. 196, 453–484 (2014). https://doi.org/10.1007/s00222-013-0472-1

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  • DOI: https://doi.org/10.1007/s00222-013-0472-1

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