{"id":"https://openalex.org/W4410702549","doi":"https://doi.org/10.1007/s10623-025-01644-6","title":"Hilbert series and degrees of regularity of Oil &amp; Vinegar and mixed quadratic systems","display_name":"Hilbert series and degrees of regularity of Oil &amp; Vinegar and mixed quadratic systems","publication_year":2025,"publication_date":"2025-05-24","ids":{"openalex":"https://openalex.org/W4410702549","doi":"https://doi.org/10.1007/s10623-025-01644-6"},"language":"en","primary_location":{"id":"doi:10.1007/s10623-025-01644-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-025-01644-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-025-01644-6.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10623-025-01644-6.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072772077","display_name":"Antonio Corbo Esposito","orcid":"https://orcid.org/0000-0003-4582-0128"},"institutions":[{"id":"https://openalex.org/I186995768","display_name":"Universit\u00e0 degli studi di Cassino e del Lazio Meridionale","ror":"https://ror.org/04nxkaq16","country_code":"IT","type":"education","lineage":["https://openalex.org/I186995768"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Antonio Corbo Esposito","raw_affiliation_strings":["Department of Electrical and Information Engineering (DIEI), University of Cassino and Southern Lazio, Via Gaetano di Biasio, 43, 03043, Cassino, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Information Engineering (DIEI), University of Cassino and Southern Lazio, Via Gaetano di Biasio, 43, 03043, Cassino, Italy","institution_ids":["https://openalex.org/I186995768"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5093402763","display_name":"Rosa Fera","orcid":null},"institutions":[{"id":"https://openalex.org/I186995768","display_name":"Universit\u00e0 degli studi di Cassino e del Lazio Meridionale","ror":"https://ror.org/04nxkaq16","country_code":"IT","type":"education","lineage":["https://openalex.org/I186995768"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Rosa Fera","raw_affiliation_strings":["Department of Electrical and Information Engineering (DIEI), University of Cassino and Southern Lazio, Via Gaetano di Biasio, 43, 03043, Cassino, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Information Engineering (DIEI), University of Cassino and Southern Lazio, Via Gaetano di Biasio, 43, 03043, Cassino, Italy","institution_ids":["https://openalex.org/I186995768"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101430324","display_name":"Francesco Romeo","orcid":"https://orcid.org/0000-0002-6176-1159"},"institutions":[{"id":"https://openalex.org/I186995768","display_name":"Universit\u00e0 degli studi di Cassino e del Lazio Meridionale","ror":"https://ror.org/04nxkaq16","country_code":"IT","type":"education","lineage":["https://openalex.org/I186995768"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Francesco Romeo","raw_affiliation_strings":["Department of Electrical and Information Engineering (DIEI), University of Cassino and Southern Lazio, Via Gaetano di Biasio, 43, 03043, Cassino, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Information Engineering (DIEI), University of Cassino and Southern Lazio, Via Gaetano di Biasio, 43, 03043, Cassino, Italy","institution_ids":["https://openalex.org/I186995768"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101430324"],"corresponding_institution_ids":["https://openalex.org/I186995768"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.11399858,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"93","issue":"9","first_page":"3521","last_page":"3541"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9997000098228455,"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":0.9997000098228455,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9886999726295471,"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"}},{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9778000116348267,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/mathematics","display_name":"Mathematics","score":0.894798994064331},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.6483911275863647},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.6053282022476196},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4130650758743286},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3423153758049011}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.894798994064331},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.6483911275863647},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.6053282022476196},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4130650758743286},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3423153758049011},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10623-025-01644-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-025-01644-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-025-01644-6.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10623-025-01644-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-025-01644-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10623-025-01644-6.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Nature"],"type":"journal"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Designs, Codes and Cryptography","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.5699999928474426,"display_name":"Zero hunger","id":"https://metadata.un.org/sdg/2"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4410702549.pdf","grobid_xml":"https://content.openalex.org/works/W4410702549.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W70420423","https://openalex.org/W575972149","https://openalex.org/W1516523879","https://openalex.org/W1518341038","https://openalex.org/W1536729852","https://openalex.org/W1548944036","https://openalex.org/W1568587963","https://openalex.org/W1569499052","https://openalex.org/W1583871653","https://openalex.org/W1618496890","https://openalex.org/W1628272610","https://openalex.org/W1821087317","https://openalex.org/W2011039300","https://openalex.org/W2067930552","https://openalex.org/W2099172492","https://openalex.org/W2117635739","https://openalex.org/W2128687423","https://openalex.org/W2128968394","https://openalex.org/W3034712312","https://openalex.org/W3095745794","https://openalex.org/W3097085459","https://openalex.org/W4281554245","https://openalex.org/W4285540167","https://openalex.org/W4386585531"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2962838298","https://openalex.org/W2600246793"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,38,49],"analyze":[4],"the":[5,41,45,54,71],"algebraic":[6],"invariants":[7],"for":[8],"two":[9],"classes":[10],"of":[11,56,73],"multivariate":[12],"quadratic":[13,21],"systems:":[14],"systems":[15,24,76],"made":[16,25],"by":[17,26],"oil":[18,28],"and":[19,23,29,32,48,60,77],"vinegar":[20,30],"polynomials":[22,31],"both":[27],"fully-quadratic":[33],"ones.":[34],"For":[35],"such":[36],"systems,":[37],"explicitly":[39],"compute":[40,70],"Hilbert":[42],"series":[43],"in":[44],"homogeneous":[46],"case,":[47],"also":[50],"give":[51],"bounds":[52],"on":[53],"degree":[55,59],"regularity,":[57],"solving":[58,74],"first":[61],"fall":[62],"degree.":[63],"Such":[64],"degrees":[65],"can":[66],"be":[67],"relevant":[68],"to":[69,78],"complexity":[72],"those":[75],"estimate":[79],"their":[80],"cryptographic":[81],"security.":[82]},"counts_by_year":[],"updated_date":"2026-06-13T06:13:01.061226","created_date":"2025-10-10T00:00:00"}
