{"id":"https://openalex.org/W2136302685","doi":"https://doi.org/10.1007/s10208-010-9062-4","title":"Polynomial Hierarchy, Betti Numbers, and\u00a0a\u00a0Real\u00a0Analogue of Toda\u2019s Theorem","display_name":"Polynomial Hierarchy, Betti Numbers, and\u00a0a\u00a0Real\u00a0Analogue of Toda\u2019s Theorem","publication_year":2010,"publication_date":"2010-02-18","ids":{"openalex":"https://openalex.org/W2136302685","doi":"https://doi.org/10.1007/s10208-010-9062-4","mag":"2136302685"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-010-9062-4","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10208-010-9062-4","pdf_url":null,"source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057052198","display_name":"Saugata Basu","orcid":"https://orcid.org/0000-0002-2441-0915"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Saugata Basu","raw_affiliation_strings":["Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA","Purdue University, Department of Mathematics, 47906, West Lafayette, IN, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, West Lafayette, IN, 47906, USA","institution_ids":["https://openalex.org/I219193219"]},{"raw_affiliation_string":"Purdue University, Department of Mathematics, 47906, West Lafayette, IN, USA#TAB#","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007037020","display_name":"Thierry Zell","orcid":null},"institutions":[{"id":"https://openalex.org/I126683464","display_name":"Lenoir\u2013Rhyne University","ror":"https://ror.org/05bggyp09","country_code":"US","type":"education","lineage":["https://openalex.org/I126683464"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Thierry Zell","raw_affiliation_strings":["School of Mathematics and Computing Sciences, Lenoir-Rhyne University, Hickory, NC, 28603, USA","Lenoir-Rhyne University, School of Mathematics and Computing Sciences, 28603, Hickory, NC, USA#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Computing Sciences, Lenoir-Rhyne University, Hickory, NC, 28603, USA","institution_ids":["https://openalex.org/I126683464"]},{"raw_affiliation_string":"Lenoir-Rhyne University, School of Mathematics and Computing Sciences, 28603, Hickory, NC, USA#TAB#","institution_ids":["https://openalex.org/I126683464"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5057052198"],"corresponding_institution_ids":["https://openalex.org/I219193219"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":null,"fwci":2.0593,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.87278851,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"10","issue":"4","first_page":"429","last_page":"454"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.991599977016449,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.9897000193595886,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7901907563209534},{"id":"https://openalex.org/keywords/complexity-class","display_name":"Complexity class","score":0.6714414954185486},{"id":"https://openalex.org/keywords/betti-number","display_name":"Betti number","score":0.5681626796722412},{"id":"https://openalex.org/keywords/polynomial-hierarchy","display_name":"Polynomial hierarchy","score":0.5618133544921875},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5090258121490479},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5051782727241516},{"id":"https://openalex.org/keywords/turing-machine","display_name":"Turing machine","score":0.4780897796154022},{"id":"https://openalex.org/keywords/counting-problem","display_name":"Counting problem","score":0.45515936613082886},{"id":"https://openalex.org/keywords/structural-complexity-theory","display_name":"Structural complexity theory","score":0.4288952946662903},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.42119234800338745},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.41751405596733093},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.4166716933250427},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4163956642150879},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.33902233839035034},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2322763204574585},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.13660115003585815},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09546676278114319}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7901907563209534},{"id":"https://openalex.org/C39637292","wikidata":"https://www.wikidata.org/wiki/Q908207","display_name":"Complexity class","level":3,"score":0.6714414954185486},{"id":"https://openalex.org/C129621563","wikidata":"https://www.wikidata.org/wiki/Q429593","display_name":"Betti number","level":2,"score":0.5681626796722412},{"id":"https://openalex.org/C2776013622","wikidata":"https://www.wikidata.org/wiki/Q2103021","display_name":"Polynomial hierarchy","level":3,"score":0.5618133544921875},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5090258121490479},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5051782727241516},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.4780897796154022},{"id":"https://openalex.org/C16592021","wikidata":"https://www.wikidata.org/wiki/Q5177154","display_name":"Counting problem","level":2,"score":0.45515936613082886},{"id":"https://openalex.org/C131583701","wikidata":"https://www.wikidata.org/wiki/Q7625020","display_name":"Structural complexity theory","level":4,"score":0.4288952946662903},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.42119234800338745},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.41751405596733093},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.4166716933250427},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4163956642150879},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.33902233839035034},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2322763204574585},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.13660115003585815},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09546676278114319},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10208-010-9062-4","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10208-010-9062-4","pdf_url":null,"source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":43,"referenced_works":["https://openalex.org/W106288804","https://openalex.org/W1511894815","https://openalex.org/W1550331971","https://openalex.org/W1553142094","https://openalex.org/W1574151794","https://openalex.org/W1589087843","https://openalex.org/W1640491905","https://openalex.org/W1849958228","https://openalex.org/W1971666500","https://openalex.org/W1980441998","https://openalex.org/W1983183758","https://openalex.org/W1991813962","https://openalex.org/W1996643891","https://openalex.org/W1999384241","https://openalex.org/W2001255903","https://openalex.org/W2001393258","https://openalex.org/W2012678207","https://openalex.org/W2014308048","https://openalex.org/W2015585803","https://openalex.org/W2057512592","https://openalex.org/W2064314361","https://openalex.org/W2065251436","https://openalex.org/W2071846104","https://openalex.org/W2082387287","https://openalex.org/W2082560626","https://openalex.org/W2082774629","https://openalex.org/W2086785654","https://openalex.org/W2087736116","https://openalex.org/W2102074804","https://openalex.org/W2115204359","https://openalex.org/W2137407595","https://openalex.org/W2143710019","https://openalex.org/W2151549243","https://openalex.org/W2152595409","https://openalex.org/W2163243521","https://openalex.org/W3102541104","https://openalex.org/W3104275909","https://openalex.org/W3107709063","https://openalex.org/W4213009134","https://openalex.org/W4230832653","https://openalex.org/W4255213956","https://openalex.org/W4298290503","https://openalex.org/W4301168154"],"related_works":["https://openalex.org/W1504200807","https://openalex.org/W2964237202","https://openalex.org/W1980428853","https://openalex.org/W2534244659","https://openalex.org/W2951395444","https://openalex.org/W4297231931","https://openalex.org/W2766363310","https://openalex.org/W2247609287","https://openalex.org/W2092011983","https://openalex.org/W2136302685"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":3}],"updated_date":"2026-05-11T08:15:01.531666","created_date":"2025-10-10T00:00:00"}