On the typical rank of real binary forms

Name: Work Video:
Duration: 3 min 30 s
Description: We determine the rank of a general real binary form of degree d=4 and d=5. In the case d=5, the possible values of the rank of such general forms are 3,4,5. The existence of three typical ranks was unexpected. We prove that a real binary form of degree d with d real roots has rank d.
Pierre Comon; Giorgio Ottaviani
{'id': 'https://openalex.org/W2952493865', 'doi': 'https://doi.org/10.48550/arxiv.0909.4865', 'title': 'On the typical rank of real binary forms', 'display_name': 'On the typical rank of real binary forms', 'publication_year': 2009, 'publication_date': '2009-01-01', 'ids': {'openalex': 'https://openalex.org/W2952493865', 'doi': 'https://doi.org/10.48550/arxiv.0909.4865', 'mag': '2952493865'}, 'language': 'en', 'primary_location': {'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/0909.4865', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, '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': ['Cornell University'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, 'type': 'preprint', 'type_crossref': 'posted-content', 'indexed_in': ['datacite'], 'open_access': {'is_oa': True, 'oa_status': 'green', 'oa_url': 'https://arxiv.org/abs/0909.4865', 'any_repository_has_fulltext': True}, 'authorships': [{'author_position': 'first', 'author': {'id': 'https://openalex.org/A5033972557', 'display_name': 'Pierre Comon', 'orcid': 'https://orcid.org/0000-0001-9436-9228'}, 'institutions': [{'id': 'https://openalex.org/I4210106479', 'display_name': "Laboratoire d'Informatique, Signaux et Systèmes de Sophia Antipolis", 'ror': 'https://ror.org/01215r597', 'country_code': 'FR', 'type': 'facility', 'lineage': ['https://openalex.org/I1294671590', 'https://openalex.org/I1326498283', 'https://openalex.org/I201841394', 'https://openalex.org/I4210106479', 'https://openalex.org/I4210159245']}], 'countries': ['FR'], 'is_corresponding': False, 'raw_author_name': 'Pierre Comon', 'raw_affiliation_strings': ["Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL"], 'affiliations': [{'raw_affiliation_string': "Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL", 'institution_ids': ['https://openalex.org/I4210106479']}]}, {'author_position': 'last', 'author': {'id': 'https://openalex.org/A5060402032', 'display_name': 'Giorgio Ottaviani', 'orcid': 'https://orcid.org/0000-0003-1275-1189'}, 'institutions': [], 'countries': [], 'is_corresponding': False, 'raw_author_name': 'Giorgio Ottaviani', 'raw_affiliation_strings': ['Dipartimento di Matematica "Ulisse Dini"'], 'affiliations': [{'raw_affiliation_string': 'Dipartimento di Matematica "Ulisse Dini"', 'institution_ids': []}]}], 'institution_assertions': [], 'countries_distinct_count': 1, 'institutions_distinct_count': 1, 'corresponding_author_ids': [], 'corresponding_institution_ids': [], 'apc_list': None, 'apc_paid': None, 'fwci': None, 'has_fulltext': False, 'cited_by_count': 9, 'citation_normalized_percentile': {'value': 0.631579, 'is_in_top_1_percent': False, 'is_in_top_10_percent': False}, 'cited_by_percentile_year': {'min': 83, 'max': 84}, 'biblio': {'volume': None, 'issue': None, 'first_page': None, 'last_page': None}, 'is_retracted': False, 'is_paratext': False, 'primary_topic': {'id': 'https://openalex.org/T12303', 'display_name': 'Tensor decomposition and applications', 'score': 0.9999, 'subfield': {'id': 'https://openalex.org/subfields/2605', 'display_name': 'Computational Mathematics'}, '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/T12303', 'display_name': 'Tensor decomposition and applications', 'score': 0.9999, 'subfield': {'id': 'https://openalex.org/subfields/2605', 'display_name': 'Computational Mathematics'}, '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/T10061', 'display_name': 'Algebraic Geometry and Number Theory', 'score': 0.9906, '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/T11130', 'display_name': 'Coding theory and cryptography', 'score': 0.9859, 'subfield': {'id': 'https://openalex.org/subfields/1702', 'display_name': 'Artificial Intelligence'}, '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/rank', 'display_name': 'Rank (graph theory)', 'score': 0.7556914}], 'concepts': [{'id': 'https://openalex.org/C164226766', 'wikidata': 'https://www.wikidata.org/wiki/Q7293202', 'display_name': 'Rank (graph theory)', 'level': 2, 'score': 0.7556914}, {'id': 'https://openalex.org/C48372109', 'wikidata': 'https://www.wikidata.org/wiki/Q3913', 'display_name': 'Binary number', 'level': 2, 'score': 0.6564871}, {'id': 'https://openalex.org/C33923547', 'wikidata': 'https://www.wikidata.org/wiki/Q395', 'display_name': 'Mathematics', 'level': 0, 'score': 0.50686437}, {'id': 'https://openalex.org/C114614502', 'wikidata': 'https://www.wikidata.org/wiki/Q76592', 'display_name': 'Combinatorics', 'level': 1, 'score': 0.36894423}, {'id': 'https://openalex.org/C94375191', 'wikidata': 'https://www.wikidata.org/wiki/Q11205', 'display_name': 'Arithmetic', 'level': 1, 'score': 0.24591333}], 'mesh': [], 'locations_count': 10, 'locations': [{'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/0909.4865', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, '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': ['Cornell University'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.archives-ouvertes.fr/hal-00420415', 'pdf_url': 'https://hal.science/hal-00420415/document', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.archives-ouvertes.fr/hal-00420415v2/file/realrank-v23.pdf', 'pdf_url': 'https://hal.archives-ouvertes.fr/hal-00420415v2/file/realrank-v23.pdf', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.archives-ouvertes.fr/hal-00700790/file/realrank-v23.pdf', 'pdf_url': 'https://hal.archives-ouvertes.fr/hal-00700790/file/realrank-v23.pdf', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.archives-ouvertes.fr/hal-00420415v2/document', 'pdf_url': 'https://hal.archives-ouvertes.fr/hal-00420415v2/document', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.archives-ouvertes.fr/hal-00700790/document', 'pdf_url': 'https://hal.archives-ouvertes.fr/hal-00700790/document', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': None, 'pdf_url': 'https://arxiv.org/pdf/0909.4865v1.pdf', 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, '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': ['Cornell University'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.science/hal-00420415v2/file/realrank-v23.pdf', 'pdf_url': 'https://hal.science/hal-00420415v2/file/realrank-v23.pdf', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': True, 'landing_page_url': 'https://hal.science/hal-00420415v2/document', 'pdf_url': 'https://hal.science/hal-00420415v2/document', 'source': {'id': 'https://openalex.org/S4306402512', 'display_name': 'HAL (Le Centre pour la Communication Scientifique Directe)', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I1294671590', 'host_organization_name': 'Centre National de la Recherche Scientifique', 'host_organization_lineage': ['https://openalex.org/I1294671590'], 'host_organization_lineage_names': ['Centre National de la Recherche Scientifique'], 'type': 'repository'}, 'license': None, 'license_id': None, 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, {'is_oa': False, 'landing_page_url': 'https://api.datacite.org/dois/10.48550/arxiv.0909.4865', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4393179698', 'display_name': 'DataCite API', 'issn_l': None, 'issn': None, 'is_oa': True, 'is_in_doaj': False, 'is_core': False, 'host_organization': 'https://openalex.org/I4210145204', 'host_organization_name': 'DataCite', 'host_organization_lineage': ['https://openalex.org/I4210145204'], 'host_organization_lineage_names': ['DataCite'], 'type': 'metadata'}, 'license': None, 'license_id': None, 'version': None}], 'best_oa_location': {'is_oa': True, 'landing_page_url': 'https://arxiv.org/abs/0909.4865', 'pdf_url': None, 'source': {'id': 'https://openalex.org/S4306400194', 'display_name': 'arXiv (Cornell University)', 'issn_l': None, 'issn': None, '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': ['Cornell University'], 'type': 'repository'}, 'license': 'other-oa', 'license_id': 'https://openalex.org/licenses/other-oa', 'version': 'submittedVersion', 'is_accepted': False, 'is_published': False}, 'sustainable_development_goals': [], 'grants': [], 'datasets': [], 'versions': [], 'referenced_works_count': 25, 'referenced_works': ['https://openalex.org/W1529395041', 'https://openalex.org/W1599617455', 'https://openalex.org/W1652996114', 'https://openalex.org/W1969432745', 'https://openalex.org/W1970866417', 'https://openalex.org/W1991386947', 'https://openalex.org/W1999538553', 'https://openalex.org/W2000215628', 'https://openalex.org/W2020062309', 'https://openalex.org/W2020465878', 'https://openalex.org/W2037271374', 'https://openalex.org/W2056790378', 'https://openalex.org/W2079407403', 'https://openalex.org/W2084586233', 'https://openalex.org/W2089007061', 'https://openalex.org/W2096735200', 'https://openalex.org/W2101641981', 'https://openalex.org/W2102950139', 'https://openalex.org/W2112835755', 'https://openalex.org/W2132267493', 'https://openalex.org/W2264065105', 'https://openalex.org/W227277429', 'https://openalex.org/W2949066980', 'https://openalex.org/W3098420274', 'https://openalex.org/W649038814'], 'related_works': ['https://openalex.org/W4391375266', 'https://openalex.org/W4238204885', 'https://openalex.org/W3002753104', 'https://openalex.org/W2600246793', 'https://openalex.org/W2142036596', 'https://openalex.org/W2077600819', 'https://openalex.org/W2072657027', 'https://openalex.org/W2061531152', 'https://openalex.org/W2007980826', 'https://openalex.org/W1979597421'], 'abstract_inverted_index': {'We': [0, 39], 'determine': [1], 'the': [2, 16, 19, 23], 'rank': [3, 24, 54], 'of': [4, 10, 22, 25, 33, 46], 'a': [5, 42], 'general': [6, 27], 'real': [7, 43, 51], 'binary': [8, 44], 'form': [9, 45], 'degree': [11, 47], 'd=4': [12], 'and': [13], 'd=5.': [14], 'In': [15], 'case': [17], 'd=5,': [18], 'possible': [20], 'values': [21], 'such': [26], 'forms': [28], 'are': [29], '3,4,5.': [30], 'The': [31], 'existence': [32], 'three': [34], 'typical': [35], 'ranks': [36], 'was': [37], 'unexpected.': [38], 'prove': [40], 'that': [41], 'd': [48, 50], 'with': [49], 'roots': [52], 'has': [53], 'd.': [55]}, 'cited_by_api_url': 'https://api.openalex.org/works?filter=cites:W2952493865', 'counts_by_year': [{'year': 2012, 'cited_by_count': 2}], 'updated_date': '2024-12-07T22:27:32.871821', 'created_date': '2019-06-27'}
Publication Information

Basic Information

Access and Citation

AI Researcher Chatbot

Primary Location

Authors

Topics

Keywords

Related Works
