{"id":"https://openalex.org/W3112488296","doi":"https://doi.org/10.1007/s10208-021-09534-8","title":"Euclidean Distance Degree and Mixed Volume","display_name":"Euclidean Distance Degree and Mixed Volume","publication_year":2021,"publication_date":"2021-09-07","ids":{"openalex":"https://openalex.org/W3112488296","doi":"https://doi.org/10.1007/s10208-021-09534-8","mag":"3112488296"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-021-09534-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09534-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09534-8.pdf","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09534-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079717788","display_name":"Paul Breiding","orcid":"https://orcid.org/0000-0003-3747-9185"},"institutions":[{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]},{"id":"https://openalex.org/I4210119712","display_name":"Max Planck Institute for Mathematics","ror":"https://ror.org/02dh8ja68","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210119712"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"P. Breiding","raw_affiliation_strings":["Max-Planck-Institute for Mathematics in the Sciences Leipzig, Leipzig, Germany"],"affiliations":[{"raw_affiliation_string":"Max-Planck-Institute for Mathematics in the Sciences Leipzig, Leipzig, Germany","institution_ids":["https://openalex.org/I4210119712","https://openalex.org/I4210091327"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5078229263","display_name":"Frank Sottile","orcid":"https://orcid.org/0000-0003-0087-7120"},"institutions":[{"id":"https://openalex.org/I91045830","display_name":"Texas A&M University","ror":"https://ror.org/01f5ytq51","country_code":"US","type":"education","lineage":["https://openalex.org/I91045830"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"F. Sottile","raw_affiliation_strings":["Department of Mathematics, Texas A&M University, College Station, Texas, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Texas A&M University, College Station, Texas, USA","institution_ids":["https://openalex.org/I91045830"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036593639","display_name":"James Woodcock","orcid":"https://orcid.org/0000-0003-4769-5375"},"institutions":[{"id":"https://openalex.org/I91045830","display_name":"Texas A&M University","ror":"https://ror.org/01f5ytq51","country_code":"US","type":"education","lineage":["https://openalex.org/I91045830"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J. Woodcock","raw_affiliation_strings":["Department of Mathematics, Texas A&M University, College Station, Texas, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Texas A&M University, College Station, Texas, USA","institution_ids":["https://openalex.org/I91045830"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5079717788"],"corresponding_institution_ids":["https://openalex.org/I4210091327","https://openalex.org/I4210119712"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.178,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.4211796,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"22","issue":"6","first_page":"1743","last_page":"1765"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9995999932289124,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9869999885559082,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/polytope","display_name":"Polytope","score":0.7897140979766846},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.7492214441299438},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.7348024845123291},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6815053224563599},{"id":"https://openalex.org/keywords/parallelepiped","display_name":"Parallelepiped","score":0.6715955138206482},{"id":"https://openalex.org/keywords/hypersurface","display_name":"Hypersurface","score":0.6476114988327026},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.6386922597885132},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.5864388942718506},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5713090896606445},{"id":"https://openalex.org/keywords/euclidean-distance-matrix","display_name":"Euclidean distance matrix","score":0.5531203746795654},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5372320413589478},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5277562737464905},{"id":"https://openalex.org/keywords/lagrange-multiplier","display_name":"Lagrange multiplier","score":0.4421055316925049},{"id":"https://openalex.org/keywords/euclidean-shortest-path","display_name":"Euclidean shortest path","score":0.43276113271713257},{"id":"https://openalex.org/keywords/volume","display_name":"Volume (thermodynamics)","score":0.42269405722618103},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.2577076554298401},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.23468557000160217},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2166961431503296},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.20630347728729248},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.10765743255615234},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08176234364509583}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.7897140979766846},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.7492214441299438},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.7348024845123291},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6815053224563599},{"id":"https://openalex.org/C2779788511","wikidata":"https://www.wikidata.org/wiki/Q181658","display_name":"Parallelepiped","level":2,"score":0.6715955138206482},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.6476114988327026},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.6386922597885132},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5864388942718506},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5713090896606445},{"id":"https://openalex.org/C171558263","wikidata":"https://www.wikidata.org/wiki/Q5406122","display_name":"Euclidean distance matrix","level":3,"score":0.5531203746795654},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5372320413589478},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5277562737464905},{"id":"https://openalex.org/C73684929","wikidata":"https://www.wikidata.org/wiki/Q598870","display_name":"Lagrange multiplier","level":2,"score":0.4421055316925049},{"id":"https://openalex.org/C170836113","wikidata":"https://www.wikidata.org/wiki/Q5406126","display_name":"Euclidean shortest path","level":5,"score":0.43276113271713257},{"id":"https://openalex.org/C20556612","wikidata":"https://www.wikidata.org/wiki/Q4469374","display_name":"Volume (thermodynamics)","level":2,"score":0.42269405722618103},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2577076554298401},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.23468557000160217},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2166961431503296},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.20630347728729248},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.10765743255615234},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08176234364509583},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C22590252","wikidata":"https://www.wikidata.org/wiki/Q1058754","display_name":"Shortest path problem","level":3,"score":0.0},{"id":"https://openalex.org/C1465435","wikidata":"https://www.wikidata.org/wiki/Q2916352","display_name":"Longest path problem","level":4,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s10208-021-09534-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09534-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09534-8.pdf","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2012.06350","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2012.06350","pdf_url":"https://arxiv.org/pdf/2012.06350","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":"","raw_type":"text"},{"id":"doi:10.48550/arxiv.2012.06350","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2012.06350","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"mag:3112488296","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.1007/s10208-021-09534-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09534-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09534-8.pdf","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2896621960","display_name":"Complexity and Condition in Algebra and Numerics","funder_award_id":"787840","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G3206327294","display_name":null,"funder_award_id":"445466444","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G3624831453","display_name":null,"funder_award_id":"Projektnummer 445466444","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G4956428346","display_name":null,"funder_award_id":"Horizon 2020 research and innovatio","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G5036817778","display_name":null,"funder_award_id":"European Union's Horizon 2020 research and innov","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G5106512922","display_name":null,"funder_award_id":"Deutsche Forschungsgemeinschaft (DFG","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G7747016121","display_name":null,"funder_award_id":"Projektnummer","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8051717526","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8318064016","display_name":null,"funder_award_id":"Horizon","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8603265992","display_name":null,"funder_award_id":"45466444","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8633428685","display_name":null,"funder_award_id":"European Union's Horizon 2020 research and innovat","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G8928656450","display_name":null,"funder_award_id":"Projekt DEAL","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320327897","display_name":"Max-Planck-Institut f\u00fcr Mathematik in den Naturwissenschaften","ror":"https://ror.org/00ez2he07"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3112488296.pdf","grobid_xml":"https://content.openalex.org/works/W3112488296.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W178772886","https://openalex.org/W219205466","https://openalex.org/W1536409494","https://openalex.org/W1570420806","https://openalex.org/W1989923040","https://openalex.org/W2007391649","https://openalex.org/W2028576930","https://openalex.org/W2037009986","https://openalex.org/W2055287661","https://openalex.org/W2089339786","https://openalex.org/W2124706092","https://openalex.org/W2151549243","https://openalex.org/W2158358539","https://openalex.org/W2158516852","https://openalex.org/W2163635171","https://openalex.org/W2318268489","https://openalex.org/W2739536107","https://openalex.org/W2784668700","https://openalex.org/W2806263336","https://openalex.org/W2963537858","https://openalex.org/W2980907638","https://openalex.org/W2988397612","https://openalex.org/W2999824720","https://openalex.org/W3057234318","https://openalex.org/W3100471031","https://openalex.org/W3103043995","https://openalex.org/W4238102028","https://openalex.org/W4244593923"],"related_works":["https://openalex.org/W3197239878","https://openalex.org/W2021262904","https://openalex.org/W2003773745","https://openalex.org/W2963095650","https://openalex.org/W2963523248","https://openalex.org/W2136264802","https://openalex.org/W2095047739","https://openalex.org/W2949380239","https://openalex.org/W1657506800","https://openalex.org/W3168378749","https://openalex.org/W2884694964","https://openalex.org/W2950973632","https://openalex.org/W2124706092","https://openalex.org/W2051359667","https://openalex.org/W2076521728","https://openalex.org/W3186803305","https://openalex.org/W41386925","https://openalex.org/W2962738276","https://openalex.org/W2951894163","https://openalex.org/W1974316204"],"abstract_inverted_index":{"Abstract":[0],"We":[1,48,79],"initiate":[2],"a":[3,19,32,91,103],"study":[4],"of":[5,13,55,69,73,83],"the":[6,11,43,46,51,66,70,74,81,94,99],"Euclidean":[7,52,95],"distance":[8,53,96],"degree":[9,54,97],"in":[10],"context":[12],"sparse":[14],"polynomials.":[15],"Specifically,":[16],"we":[17],"consider":[18],"hypersurface":[20],"$$f=0$$":[21,56],"<mml:math":[22,57],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[23,58],"<mml:mrow>":[24,59],"<mml:mi>f</mml:mi>":[25,60],"<mml:mo>=</mml:mo>":[26,61],"<mml:mn>0</mml:mn>":[27,62],"</mml:mrow>":[28,63],"</mml:math>":[29,64],"defined":[30],"by":[31],"polynomial":[33],"f":[34],"that":[35,42,50],"is":[36,102],"general":[37],"given":[38],"its":[39],"support,":[40],"such":[41],"support":[44],"contains":[45],"origin.":[47],"show":[49],"equals":[65],"mixed":[67],"volume":[68],"Newton":[71,100],"polytopes":[72],"associated":[75],"Lagrange":[76],"multiplier":[77],"equations.":[78],"discuss":[80],"implication":[82],"our":[84],"result":[85],"for":[86,93],"computational":[87],"complexity":[88],"and":[89],"give":[90],"formula":[92],"when":[98],"polytope":[101],"rectangular":[104],"parallelepiped.":[105]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-13T07:58:08.660418","created_date":"2025-10-10T00:00:00"}
