{"id":"https://openalex.org/W2913056486","doi":"https://doi.org/10.1007/s00454-020-00246-4","title":"Classification of Triples of Lattice Polytopes with a Given Mixed Volume","display_name":"Classification of Triples of Lattice Polytopes with a Given Mixed Volume","publication_year":2020,"publication_date":"2020-10-13","ids":{"openalex":"https://openalex.org/W2913056486","doi":"https://doi.org/10.1007/s00454-020-00246-4","mag":"2913056486"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-020-00246-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00246-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00246-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00246-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020875207","display_name":"Gennadiy Averkov","orcid":"https://orcid.org/0000-0003-2245-9958"},"institutions":[{"id":"https://openalex.org/I51783024","display_name":"Brandenburg University of Technology Cottbus-Senftenberg","ror":"https://ror.org/02wxx3e24","country_code":"DE","type":"education","lineage":["https://openalex.org/I51783024"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Gennadiy Averkov","raw_affiliation_strings":["Fakult\u00e4t 1, BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Fakult\u00e4t 1, BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Germany","institution_ids":["https://openalex.org/I51783024"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037762248","display_name":"Christopher Borger","orcid":"https://orcid.org/0000-0002-9735-394X"},"institutions":[{"id":"https://openalex.org/I95793202","display_name":"Otto-von-Guericke-Universit\u00e4t Magdeburg","ror":"https://ror.org/00ggpsq73","country_code":"DE","type":"education","lineage":["https://openalex.org/I95793202"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Christopher Borger","raw_affiliation_strings":["Faculty of Mathematics, Otto-von-Guericke-Universit\u00e4t Magdeburg, Universit\u00e4tsplatz 2, 39106, Magdeburg, Germany"],"raw_orcid":"https://orcid.org/0000-0002-9735-394X","affiliations":[{"raw_affiliation_string":"Faculty of Mathematics, Otto-von-Guericke-Universit\u00e4t Magdeburg, Universit\u00e4tsplatz 2, 39106, Magdeburg, Germany","institution_ids":["https://openalex.org/I95793202"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059943366","display_name":"Ivan Soprunov","orcid":"https://orcid.org/0000-0003-0122-7699"},"institutions":[{"id":"https://openalex.org/I102607778","display_name":"Cleveland State University","ror":"https://ror.org/002tx1f22","country_code":"US","type":"education","lineage":["https://openalex.org/I102607778"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ivan Soprunov","raw_affiliation_strings":["Department of Mathematics and Statistics, Cleveland State University, 2121 Euclid Ave, Cleveland, OH, 44115, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Cleveland State University, 2121 Euclid Ave, Cleveland, OH, 44115, USA","institution_ids":["https://openalex.org/I102607778"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5037762248"],"corresponding_institution_ids":["https://openalex.org/I95793202"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":3.2874,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.91133005,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"66","issue":"1","first_page":"165","last_page":"202"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9990000128746033,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.9065805673599243},{"id":"https://openalex.org/keywords/enumeration","display_name":"Enumeration","score":0.8148269057273865},{"id":"https://openalex.org/keywords/monomial","display_name":"Monomial","score":0.811650812625885},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8034036159515381},{"id":"https://openalex.org/keywords/mixed-volume","display_name":"Mixed volume","score":0.761297345161438},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6331236362457275},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.6166456937789917},{"id":"https://openalex.org/keywords/complex-torus","display_name":"Complex torus","score":0.5291308164596558},{"id":"https://openalex.org/keywords/torus","display_name":"Torus","score":0.4633370637893677},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4375148415565491},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.43725132942199707},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3934628665447235},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0852895975112915},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07740211486816406},{"id":"https://openalex.org/keywords/convex-body","display_name":"Convex body","score":0.07664686441421509}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.9065805673599243},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.8148269057273865},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.811650812625885},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8034036159515381},{"id":"https://openalex.org/C44550424","wikidata":"https://www.wikidata.org/wiki/Q4424496","display_name":"Mixed volume","level":5,"score":0.761297345161438},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6331236362457275},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.6166456937789917},{"id":"https://openalex.org/C28890030","wikidata":"https://www.wikidata.org/wiki/Q5156610","display_name":"Complex torus","level":3,"score":0.5291308164596558},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.4633370637893677},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4375148415565491},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.43725132942199707},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3934628665447235},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0852895975112915},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07740211486816406},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.07664686441421509},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.0},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s00454-020-00246-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00246-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00246-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1902.00891","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1902.00891","pdf_url":"https://arxiv.org/pdf/1902.00891","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:engagedscholarship.csuohio.edu:scimath_facpub-1342","is_oa":true,"landing_page_url":"https://engagedscholarship.csuohio.edu/scimath_facpub/338","pdf_url":null,"source":{"id":"https://openalex.org/S4306400889","display_name":"EngagedScholarship @ Cleveland State University (Cleveland State University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I102607778","host_organization_name":"Cleveland State University","host_organization_lineage":["https://openalex.org/I102607778"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics Faculty Publications","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1007/s00454-020-00246-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-020-00246-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-020-00246-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.550000011920929,"display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G3150829247","display_name":null,"funder_award_id":"314838170","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2913056486.pdf","grobid_xml":"https://content.openalex.org/works/W2913056486.grobid-xml"},"referenced_works_count":46,"referenced_works":["https://openalex.org/W1483218536","https://openalex.org/W1536409494","https://openalex.org/W1539239987","https://openalex.org/W1552211935","https://openalex.org/W1571966798","https://openalex.org/W1636602783","https://openalex.org/W1669332885","https://openalex.org/W1885919460","https://openalex.org/W1937803686","https://openalex.org/W1963534149","https://openalex.org/W1963547452","https://openalex.org/W1976677460","https://openalex.org/W1990510764","https://openalex.org/W2014380411","https://openalex.org/W2015438979","https://openalex.org/W2018441586","https://openalex.org/W2021567595","https://openalex.org/W2143023483","https://openalex.org/W2573816347","https://openalex.org/W2626020844","https://openalex.org/W2753535706","https://openalex.org/W2784668700","https://openalex.org/W2883073730","https://openalex.org/W2913056486","https://openalex.org/W2927622555","https://openalex.org/W2953049358","https://openalex.org/W2963553657","https://openalex.org/W2963917820","https://openalex.org/W2964281192","https://openalex.org/W2969452531","https://openalex.org/W3005385089","https://openalex.org/W3011847229","https://openalex.org/W3100261291","https://openalex.org/W3102349276","https://openalex.org/W3102864083","https://openalex.org/W3105047929","https://openalex.org/W3105369342","https://openalex.org/W3105921057","https://openalex.org/W3158465085","https://openalex.org/W4237179050","https://openalex.org/W4237930986","https://openalex.org/W4290464992","https://openalex.org/W4292096448","https://openalex.org/W4298197110","https://openalex.org/W4301808069","https://openalex.org/W6604764851"],"related_works":["https://openalex.org/W108845024","https://openalex.org/W4300125793","https://openalex.org/W180925886","https://openalex.org/W4302565374","https://openalex.org/W2773081500","https://openalex.org/W3092837321","https://openalex.org/W4293281679","https://openalex.org/W4384575929","https://openalex.org/W4280542881","https://openalex.org/W2913056486"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"present":[2],"an":[3],"algorithm":[4],"for":[5,44],"the":[6,26,32,38,82],"classification":[7,27,69],"of":[8,10,34,40,56,70,89,95,102],"triples":[9,55],"lattice":[11],"polytopes":[12],"with":[13,76],"a":[14,68,92,100],"given":[15],"mixed":[16,58],"volume":[17,59],"m":[18,46],"in":[19,81],"dimension":[20],"3.":[21],"It":[22],"is":[23,42],"known":[24],"that":[25,63,109],"can":[28],"be":[29],"reduced":[30],"to":[31,61,78,86,99],"enumeration":[33],"so-called":[35],"irreducible":[36,54],"triples,":[37],"number":[39],"which":[41],"finite":[43],"fixed":[45],".":[47],"Following":[48],"this":[49,97],"algorithm,":[50],"we":[51],"enumerate":[52],"all":[53,103],"normalized":[57],"up":[60,77,85],"4":[62,79],"are":[64,110],"inclusion-maximal.":[65],"This":[66],"produces":[67],"generic":[71,104],"trivariate":[72,105],"sparse":[73,106],"polynomial":[74,107],"systems":[75,108],"solutions":[80],"complex":[83],"torus,":[84],"monomial":[87],"changes":[88],"variables.":[90],"By":[91],"recent":[93],"result":[94],"Esterov,":[96],"leads":[98],"description":[101],"solvable":[111],"by":[112],"radicals.":[113]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}