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Primitive Recursive Ordered Fields and Some Applications

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Computer Algebra in Scientific Computing (CASC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12865))

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Abstract

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals and apply them to several problems of algebra and analysis. In particular, we find a primitive recursive analogue of Ershov-Madison’s theorem about the computable real closure, relate primitive recursive fields of reals to the field of primitive recursive reals, give sufficient conditions for primitive recursive root-finding and for computing solution operators of symmetric hyperbolic systems of partial differential equations.

V. Selivanov—The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.

S. Selivanova—The work is partially supported by RFBR-JSPS Grant 20-51-50001, by the National Research Foundation of Korea (grant 2017R1E1A1A03071032), by the International Research & Development Program of the Korean Ministry of Science and ICT (grant 2016K1A3A7A03950702), and by the NRF Brain Pool program (grant 2019H1D3A2A02102240).

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Notes

  1. 1.

    We are grateful to Vasco Brattka for the hint to Goodstein’s monography.

  2. 2.

    We are grateful to an anonymous reviewer for the hint to the survey by W. Gomaa.

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Acknowledgments

The authors thank Pavel Alaev, Sergey Goncharov, Valentina Harizanov, Peter Hertling, Iskander Kalimullin, Julia Knight, Russell Miller and Andrey Morozov for useful discussions. The first author is grateful to Arcadia University and Xizhong Zheng for the hospitality, support, and useful discussions.

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Authors and Affiliations

  1. A. P. Ershov Institute of Informatics Systems, and S. L. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Victor Selivanov

  2. KAIST, School of Computing, Daejeon, Republic of Korea

    Svetlana Selivanova

Authors
  1. Victor Selivanov
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  2. Svetlana Selivanova
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Correspondence to Victor Selivanov .

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Editors and Affiliations

  1. Université de Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Selivanov, V., Selivanova, S. (2021). Primitive Recursive Ordered Fields and Some Applications. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2021. Lecture Notes in Computer Science(), vol 12865. Springer, Cham. https://doi.org/10.1007/978-3-030-85165-1_20

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  • DOI: https://doi.org/10.1007/978-3-030-85165-1_20

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