{"id":"https://openalex.org/W7215810055","doi":"https://doi.org/10.48550/arxiv.2610.02177","title":"Generic solutions to symmetric linear equations","display_name":"Generic solutions to symmetric linear equations","publication_year":2026,"publication_date":"2026-10-01","ids":{"openalex":"https://openalex.org/W7215810055","doi":"https://doi.org/10.48550/arxiv.2610.02177"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.02177","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02177","pdf_url":"https://arxiv.org/pdf/2610.02177","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.02177","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015840899","display_name":"Bryce Frederickson","orcid":"https://orcid.org/0000-0002-5983-1701","observed_orcids":["https://orcid.org/0000-0002-5983-1701"]},"institutions":[{"id":"https://openalex.org/I150468666","display_name":"Emory University","ror":"https://ror.org/03czfpz43","country_code":"US","type":"education","lineage":["https://openalex.org/I150468666"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Frederickson, Bryce","raw_affiliation_strings":["Department of Mathematics , Emory University , Atlanta , GA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Emory University , Atlanta , GA , USA","institution_ids":["https://openalex.org/I150468666"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066277903","display_name":"Liana Yepremyan","orcid":"https://orcid.org/0000-0001-7277-4273","observed_orcids":["https://orcid.org/0000-0001-7277-4273"]},"institutions":[{"id":"https://openalex.org/I150468666","display_name":"Emory University","ror":"https://ror.org/03czfpz43","country_code":"US","type":"education","lineage":["https://openalex.org/I150468666"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yepremyan, Liana","raw_affiliation_strings":["Department of Mathematics , Emory University , Atlanta , GA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Emory University , Atlanta , GA , USA","institution_ids":["https://openalex.org/I150468666"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I150468666"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9926999807357788,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9926999807357788,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.005100000184029341,"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/T11166","display_name":"Analytic Number Theory Research","score":0.0013000000035390258,"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/subset-sums","display_name":"subset sums","score":0.8349999785423279},{"id":"https://openalex.org/keywords/supersaturation","display_name":"supersaturation","score":0.7860000133514404},{"id":"https://openalex.org/keywords/finite-abelian-groups","display_name":"finite abelian groups","score":0.7329999804496765},{"id":"https://openalex.org/keywords/even-cycles","display_name":"even cycles","score":0.609000027179718},{"id":"https://openalex.org/keywords/vector-spaces-over-finite-fields","display_name":"vector spaces over finite fields","score":0.671999990940094},{"id":"https://openalex.org/keywords/additive-combinatorics","display_name":"additive combinatorics","score":0.7059999704360962}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8062000274658203},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.7023000121116638},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.6083999872207642},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4514999985694885},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3935000002384186},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.38909998536109924},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.3587999939918518},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3546999990940094},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.334199994802475},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.33309999108314514},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.3296000063419342},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3292999863624573},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C2779832538","wikidata":"https://www.wikidata.org/wiki/Q2308809","display_name":"Coincidence","level":3,"score":0.3009999990463257},{"id":"https://openalex.org/C106286213","wikidata":"https://www.wikidata.org/wiki/Q898572","display_name":"Matroid","level":2,"score":0.30059999227523804},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.29679998755455017},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.29670000076293945}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.02177","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02177","pdf_url":"https://arxiv.org/pdf/2610.02177","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2610.02177","doi":"https://doi.org/10.48550/arxiv.2610.02177","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02177","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.02177","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02177","pdf_url":"https://arxiv.org/pdf/2610.02177","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1020685157","display_name":"Forbidden and Colored Subgraphs","funder_award_id":"2247013","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215810055.pdf","grobid_xml":"https://content.openalex.org/works/W7215810055.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W1709693601","https://openalex.org/W3195905058","https://openalex.org/W4396243460","https://openalex.org/W4401554793","https://openalex.org/W4404990932","https://openalex.org/W4415462986","https://openalex.org/W7169587086"],"related_works":["https://openalex.org/W2027904717","https://openalex.org/W2110985552","https://openalex.org/W7100757077","https://openalex.org/W2312854485","https://openalex.org/W3104730994","https://openalex.org/W2089905104","https://openalex.org/W2194022954","https://openalex.org/W2146275066","https://openalex.org/W2130495610","https://openalex.org/W7221537167"],"abstract_inverted_index":{"In":[0],"1993,":[1],"Ruzsa":[2],"showed":[3],"that":[4,16,41,59,76,93,125],"for":[5,136,154,208,229],"every":[6,17],"$k":[7,231],"\\geq":[8,128,232],"2$,":[9],"there":[10,130],"exists":[11],"a":[12,121,150,245],"constant":[13,210,227],"$C$":[14,211,228],"such":[15,40],"subset":[18,86],"$A":[19],"\\subseteq":[20],"[N]$":[21],"of":[22,114,157,176,184,198,248],"size":[23,199,203],"at":[24,132,204],"least":[25,133],"$C":[26,206],"N^{1/k}$":[27],"contains":[28],"$2k$":[29,201],"distinct":[30,85],"elements":[31,61],"$a_1,":[32,62,137],"\\ldots,":[33,36,63,66,78,81,95,99,138,141],"a_k,":[34,64,79,139],"b_1,":[35,65,80,140],"b_k":[37,142],"\\in":[38,143],"A$":[39,144],"$a_1":[42],"+":[43,45,49,51],"\\cdots":[44,50],"a_k":[46],"=":[47],"b_1":[48],"b_k$.":[52],"We":[53,148,160],"strengthen":[54],"this":[55,220],"result":[56],"by":[57],"proving":[58],"the":[60,73,89,102,166,172,177,181,226,239,249],"b_k$":[67],"can":[68],"be":[69],"chosen":[70],"to":[71,109,165,225,238],"have":[72,101],"additional":[74],"property":[75],"$\\{a_1,":[77,94],"b_k\\}$":[82,100],"has":[83,202],"$2^{2k}-1$":[84],"sums,":[87],"with":[88,195],"only":[90,213],"coincidence":[91],"being":[92],"a_k\\}$":[96],"and":[97,118,170,216,243,252],"$\\{b_1,":[98],"same":[103],"sum.":[104],"Our":[105,234],"proof":[106,247],"also":[107,161,236],"applies":[108],"any":[110],"finite":[111],"Abelian":[112,155],"group":[113],"odd":[115],"order":[116],"$N$,":[117],"it":[119],"provides":[120],"corresponding":[122],"supersaturation":[123,254],"result:":[124],"whenever":[126],"$|A|":[127],"CN^{1/k}$,":[129],"are":[131],"$\u03a9(|A|^{2k}/N)$":[134],"choices":[135],"satisfying":[145],"these":[146],"properties.":[147],"prove":[149,171],"slightly":[151],"weaker":[152],"statement":[153],"groups":[156],"even":[158,185],"order.":[159],"apply":[162,237],"our":[163],"methods":[164,235],"vector":[167],"space":[168],"setting":[169,242],"following":[173],"$\\mathbb":[174,192],"F_q$-analogue":[175],"Bondy-Simonovits":[178,250],"Theorem":[179,251],"on":[180,214],"extremal":[182],"number":[183],"cycles":[186],"in":[187],"graphs:":[188],"Any":[189],"rank-$n$,":[190],"simple,":[191],"F_q$-representable":[193],"matroid":[194],"no":[196],"circuit":[197],"exactly":[200],"most":[205],"q^{n/k}$":[207],"some":[209],"depending":[212],"$q$":[215],"$k$.":[217],"When":[218],"$q=2$,":[219],"is":[221],"best":[222],"possible":[223],"up":[224],"all":[230],"2$.":[233],"original":[240],"graph":[241],"give":[244],"new":[246],"its":[253],"version.":[255]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-03T00:00:00"}