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Polynomial-Time Presentations of Algebraic Number Fields

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Book cover Sailing Routes in the World of Computation (CiE 2018)

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Abstract

Using an extension of the notion of polynomial time presentable structure we show that some natural presentations of the ordered field \({\mathbb R}_{\text{ alg }}\) of algebraic reals and of the field \({\mathbb C}_{\text{ alg }}\) of algebraic complex numbers are polynomial-time equivalent to each other and are polynomial time. We also establish upper complexity bounds for the problem of rational polynomial evaluation in \({\mathbb C}_{\text{ alg }}\) and for the problem of root-finding for polynomials in \({\mathbb C}_{\text{ alg }}[x]\) which improve the previously known bound.

P. Alaev—The work of first author was funded by RFBR, the research project 17-01-00247.

V. Selivanov—The work of second author was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project No 1.12878.2018/12.1.

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Author information

Authors and Affiliations

  1. S. L. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Pavel Alaev

  2. Novosibirsk State University, Novosibirsk, Russia

    Pavel Alaev

  3. A. P. Ershov Institute of Informatics Systems SB RAS, Novosibirsk, Russia

    Victor Selivanov

  4. Kazan Federal University, Kazan, Russia

    Victor Selivanov

Authors
  1. Pavel Alaev
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  2. Victor Selivanov
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Correspondence to Pavel Alaev .

Editor information

Editors and Affiliations

  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

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Alaev, P., Selivanov, V. (2018). Polynomial-Time Presentations of Algebraic Number Fields. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_2

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  • DOI: https://doi.org/10.1007/978-3-319-94418-0_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94417-3

  • Online ISBN: 978-3-319-94418-0

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