{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72543"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72543","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Maximum Betti Numbers for a Given Hilbert Function","abstract":"In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\\lbrack x\\sb1,\\..., x\\sb{n}\\rbrack$ where k is a field of characteristic 0. He constructed the lex-segment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lex-segment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lex-segment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above.","abstract_html":"In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\\lbrack x\\sb1,\\..., x\\sb{n}\\rbrack$ where k is a field of characteristic 0. He constructed the lex-segment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lex-segment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lex-segment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above.","abstract_has_math":true,"creators":["Hulett, Heather Ann"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Evans, E.G.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-17T23:17:48Z","date_published":"2014-12-17T23:17:48Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9411657"],"render_values":[{"text":"(UMI)AAI9411657","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72543","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Evans, E.G.,"]},{"key":"dc:creator","label":"Author","values":["Hulett, Heather Ann"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:48Z","10000-01-01","1993"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72543","(UMI)AAI9411657"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\\lbrack x\\sb1,\\..., x\\sb{n}\\rbrack$ where k is a field of characteristic 0. He constructed the lex-segment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lex-segment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lex-segment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above.","Made available in DSpace on 2014-12-17T23:17:48Z (GMT). No. of bitstreams: 1 9411657.pdf: 2464234 bytes, checksum: 2c21aaa38722a2dbe19eb8209fb1a9d1 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72711 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","60 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."]},{"key":"dc:title","label":"Title","values":["Maximum Betti Numbers for a Given Hilbert Function"]}]}],"canonical_facts":{"dc:contributor":["Evans, E.G.,"],"dc:creator":["Hulett, Heather Ann"],"dc:date":["2014-12-17T23:17:48Z","10000-01-01","1993"],"dc:description":["In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\\lbrack x\\sb1,\\..., x\\sb{n}\\rbrack$ where k is a field of characteristic 0. He constructed the lex-segment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lex-segment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lex-segment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above.","Made available in DSpace on 2014-12-17T23:17:48Z (GMT). No. of bitstreams: 1 9411657.pdf: 2464234 bytes, checksum: 2c21aaa38722a2dbe19eb8209fb1a9d1 (MD5) Previous issue date: 1993","Embargo set by: Seth Robbins for item 72711 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","60 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993."],"dc:identifier":["http://hdl.handle.net/2142/72543","(UMI)AAI9411657"],"dc:subject":["Mathematics"],"dc:title":["Maximum Betti Numbers for a Given Hilbert Function"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:07Z"}