Fields of Algebraic Numbers Computable in Polynomial Time. II

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 i...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Alaev, P. E. [verfasserIn] Selivanov, V. L.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	field of algebraic numbers polynomial-time computable structures equivalence of polynomial-time computable structures

Anmerkung:	© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Übergeordnetes Werk:	Enthalten in: Algebra and logic - Dordrecht [u.a.] : Springer Science + Business Media B.V, 1968, 60(2022), 6 vom: Jan., Seite 349-359
Übergeordnetes Werk:	volume:60 ; year:2022 ; number:6 ; month:01 ; pages:349-359

Links:	Volltext

DOI / URN:	10.1007/s10469-022-09661-3

Katalog-ID:	SPR047242752

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allfields	10.1007/s10469-022-09661-3 doi (DE-627)SPR047242752 (SPR)s10469-022-09661-3-e DE-627 ger DE-627 rakwb eng Alaev, P. E. verfasserin aut Fields of Algebraic Numbers Computable in Polynomial Time. II 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 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. field of algebraic numbers (dpeaa)DE-He213 polynomial-time computable structures (dpeaa)DE-He213 equivalence of polynomial-time computable structures (dpeaa)DE-He213 Selivanov, V. L. aut Enthalten in Algebra and logic Dordrecht [u.a.] : Springer Science + Business Media B.V, 1968 60(2022), 6 vom: Jan., Seite 349-359 (DE-627)325568294 (DE-600)2037101-9 1573-8302 nnns volume:60 year:2022 number:6 month:01 pages:349-359 https://dx.doi.org/10.1007/s10469-022-09661-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 60 2022 6 01 349-359
spelling	10.1007/s10469-022-09661-3 doi (DE-627)SPR047242752 (SPR)s10469-022-09661-3-e DE-627 ger DE-627 rakwb eng Alaev, P. E. verfasserin aut Fields of Algebraic Numbers Computable in Polynomial Time. II 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 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. field of algebraic numbers (dpeaa)DE-He213 polynomial-time computable structures (dpeaa)DE-He213 equivalence of polynomial-time computable structures (dpeaa)DE-He213 Selivanov, V. L. aut Enthalten in Algebra and logic Dordrecht [u.a.] : Springer Science + Business Media B.V, 1968 60(2022), 6 vom: Jan., Seite 349-359 (DE-627)325568294 (DE-600)2037101-9 1573-8302 nnns volume:60 year:2022 number:6 month:01 pages:349-359 https://dx.doi.org/10.1007/s10469-022-09661-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 60 2022 6 01 349-359
allfields_unstemmed	10.1007/s10469-022-09661-3 doi (DE-627)SPR047242752 (SPR)s10469-022-09661-3-e DE-627 ger DE-627 rakwb eng Alaev, P. E. verfasserin aut Fields of Algebraic Numbers Computable in Polynomial Time. II 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 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. field of algebraic numbers (dpeaa)DE-He213 polynomial-time computable structures (dpeaa)DE-He213 equivalence of polynomial-time computable structures (dpeaa)DE-He213 Selivanov, V. L. aut Enthalten in Algebra and logic Dordrecht [u.a.] : Springer Science + Business Media B.V, 1968 60(2022), 6 vom: Jan., Seite 349-359 (DE-627)325568294 (DE-600)2037101-9 1573-8302 nnns volume:60 year:2022 number:6 month:01 pages:349-359 https://dx.doi.org/10.1007/s10469-022-09661-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 60 2022 6 01 349-359
allfieldsGer	10.1007/s10469-022-09661-3 doi (DE-627)SPR047242752 (SPR)s10469-022-09661-3-e DE-627 ger DE-627 rakwb eng Alaev, P. E. verfasserin aut Fields of Algebraic Numbers Computable in Polynomial Time. II 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 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. field of algebraic numbers (dpeaa)DE-He213 polynomial-time computable structures (dpeaa)DE-He213 equivalence of polynomial-time computable structures (dpeaa)DE-He213 Selivanov, V. L. aut Enthalten in Algebra and logic Dordrecht [u.a.] : Springer Science + Business Media B.V, 1968 60(2022), 6 vom: Jan., Seite 349-359 (DE-627)325568294 (DE-600)2037101-9 1573-8302 nnns volume:60 year:2022 number:6 month:01 pages:349-359 https://dx.doi.org/10.1007/s10469-022-09661-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 60 2022 6 01 349-359
allfieldsSound	10.1007/s10469-022-09661-3 doi (DE-627)SPR047242752 (SPR)s10469-022-09661-3-e DE-627 ger DE-627 rakwb eng Alaev, P. E. verfasserin aut Fields of Algebraic Numbers Computable in Polynomial Time. II 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 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. field of algebraic numbers (dpeaa)DE-He213 polynomial-time computable structures (dpeaa)DE-He213 equivalence of polynomial-time computable structures (dpeaa)DE-He213 Selivanov, V. L. aut Enthalten in Algebra and logic Dordrecht [u.a.] : Springer Science + Business Media B.V, 1968 60(2022), 6 vom: Jan., Seite 349-359 (DE-627)325568294 (DE-600)2037101-9 1573-8302 nnns volume:60 year:2022 number:6 month:01 pages:349-359 https://dx.doi.org/10.1007/s10469-022-09661-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 60 2022 6 01 349-359
language	English
source	Enthalten in Algebra and logic 60(2022), 6 vom: Jan., Seite 349-359 volume:60 year:2022 number:6 month:01 pages:349-359
sourceStr	Enthalten in Algebra and logic 60(2022), 6 vom: Jan., Seite 349-359 volume:60 year:2022 number:6 month:01 pages:349-359
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topic_facet	field of algebraic numbers polynomial-time computable structures equivalence of polynomial-time computable structures
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container_title	Algebra and logic
authorswithroles_txt_mv	Alaev, P. E. @@aut@@ Selivanov, V. L. @@aut@@
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author	Alaev, P. E.
spellingShingle	Alaev, P. E. misc field of algebraic numbers misc polynomial-time computable structures misc equivalence of polynomial-time computable structures Fields of Algebraic Numbers Computable in Polynomial Time. II
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topic_title	Fields of Algebraic Numbers Computable in Polynomial Time. II field of algebraic numbers (dpeaa)DE-He213 polynomial-time computable structures (dpeaa)DE-He213 equivalence of polynomial-time computable structures (dpeaa)DE-He213
topic	misc field of algebraic numbers misc polynomial-time computable structures misc equivalence of polynomial-time computable structures
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title	Fields of Algebraic Numbers Computable in Polynomial Time. II
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title_full	Fields of Algebraic Numbers Computable in Polynomial Time. II
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doi_str_mv	10.1007/s10469-022-09661-3
title_sort	fields of algebraic numbers computable in polynomial time. ii
title_auth	Fields of Algebraic Numbers Computable in Polynomial Time. II
abstract	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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
abstractGer	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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
abstract_unstemmed	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. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
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title_short	Fields of Algebraic Numbers Computable in Polynomial Time. II
url	https://dx.doi.org/10.1007/s10469-022-09661-3
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II</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2022</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">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. 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