This paper is a continuation of [Algebra and Logic, 58, No. 6, 447-469 (2019)] where we constructed polynomial-time presentations for the field of complex algebraic numbers and for the ordered field of real algebraic numbers. Here we discuss other known natural presentations of such structures. It is shown that all these presentations are equivalent to each other and prove a theorem which explains why this is so. While analyzing the presentations mentioned, we introduce the notion of a quotient structure. It is shown that the question whether a polynomial-time computable quotient structure is equivalent to an ordinary one is almost equivalent to the P = NP problem. Conditions are found under which the answer is positive.
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Translated from Algebra i Logika, Vol. 60, No. 6, pp. 533-548, November-December, 2021. Russian DOI: https://doi.org/10.33048/alglog.2021.60.601
Supported by Mathematical Center in Akademgorodok, Agreement with RF Ministry of Education and Science No. 075-15-2019-1613.
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Alaev, P.E., Selivanov, V.L. Fields of Algebraic Numbers Computable in Polynomial Time. II. Algebra Logic 60, 349–359 (2022). https://doi.org/10.1007/s10469-022-09661-3
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DOI: https://doi.org/10.1007/s10469-022-09661-3