{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72534"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72534","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Real Even Symmetric Forms","abstract":"Let F$\\sb{n,m}$ denote the set of all real forms of degree m in n variables. In 1888, Hilbert proved that a form P $\\in$ F$\\sb{n,m}$ which is positive semidefinite (psd) must have a representation as a sum of squares (sos) of forms if and only if n = 2, m = 2, or (n,m) = (3,4). No concrete example of a psd form which is not sos was known until the late 1960's. We denote by S$\\sbsp{n,m}{e}$ the set of all real symmetric forms of degree m = 2d. Let PS$\\sbsp{n,m}{e}$ and $\\Sigma$S$\\sbsp{n,m}{e}$ denote the cones of psd and sos elements of S$\\sbsp{n,m}{e},$ respectively. For m = 2 or 4, these cones coincide. For m = 6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam and B. Reznick, Math. Z. 195 (1987), pp. 559-580.","abstract_html":"Let F$\\sb{n,m}$ denote the set of all real forms of degree m in n variables. In 1888, Hilbert proved that a form P $\\in$ F$\\sb{n,m}$ which is positive semidefinite (psd) must have a representation as a sum of squares (sos) of forms if and only if n = 2, m = 2, or (n,m) = (3,4). No concrete example of a psd form which is not sos was known until the late 1960's. We denote by S$\\sbsp{n,m}{e}$ the set of all real symmetric forms of degree m = 2d. Let PS$\\sbsp{n,m}{e}$ and $\\Sigma$S$\\sbsp{n,m}{e}$ denote the cones of psd and sos elements of S$\\sbsp{n,m}{e},$ respectively. For m = 2 or 4, these cones coincide. For m = 6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam and B. Reznick, Math. Z. 195 (1987), pp. 559-580.","abstract_has_math":true,"creators":["Harris, William Richard"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Reznick, B.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-17T23:17:45Z","date_published":"2014-12-17T23:17:45Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9305548"],"render_values":[{"text":"(UMI)AAI9305548","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72534","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Reznick, B.,"]},{"key":"dc:creator","label":"Author","values":["Harris, William Richard"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:45Z","10000-01-01","1992"]},{"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/2142/72534","(UMI)AAI9305548"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let F$\\sb{n,m}$ denote the set of all real forms of degree m in n variables. In 1888, Hilbert proved that a form P $\\in$ F$\\sb{n,m}$ which is positive semidefinite (psd) must have a representation as a sum of squares (sos) of forms if and only if n = 2, m = 2, or (n,m) = (3,4). No concrete example of a psd form which is not sos was known until the late 1960's. We denote by S$\\sbsp{n,m}{e}$ the set of all real symmetric forms of degree m = 2d. Let PS$\\sbsp{n,m}{e}$ and $\\Sigma$S$\\sbsp{n,m}{e}$ denote the cones of psd and sos elements of S$\\sbsp{n,m}{e},$ respectively. For m = 2 or 4, these cones coincide. For m = 6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam and B. Reznick, Math. Z. 195 (1987), pp. 559-580.","We present an easily-checked, necessary and sufficient condition for an even symmetric n-ary octic to be in PS$\\sbsp{n,8}{e}$ and for an even symmetric ternary decic to be in PS$\\sbsp{3,10}{e},$ and also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10. We proceed to discuss the extremal elements of the cones PS$\\sbsp{3,8}{e},$ PS$\\sbsp{3,10}{e}$ and PS$\\sbsp{4,8}{e}.$ This leads to the question: how many of these extremal forms have sos representations? We prove that PS$\\sbsp{3,8}{e}$ = $\\Sigma$S$\\sbsp{3,8}{e},$ a companion result to Hilbert's theorem noted above, with regard to psd ternary quartics. We also demonstrate that neither PS$\\sbsp{3,10}{e}\\\\\\Sigma$S$\\sbsp{3,10}{e}$ nor PS$\\sbsp{4,8}{e}\\\\\\Sigma$S$\\sbsp{4,8}{e}$ is empty, providing many new examples of psd forms which are not sos.","We give a graphic representation with examples of ternary forms which also indicates whether or not an element of S$\\sbsp{3,8}{e}$ or S$\\sbsp{3,10}{e}$ is psd. We interpret elements of PS$\\sbsp{n,m}{e}$ as inequalities; in particular, we give all symmetric polynomial inequalities of degree $\\le$5 satisfied by the sides of a triangle.","Made available in DSpace on 2014-12-17T23:17:45Z (GMT). No. of bitstreams: 1 9305548.pdf: 3598174 bytes, checksum: afd93a2944786588bcf3375d5029a0a6 (MD5) Previous issue date: 1992","Embargo set by: Seth Robbins for item 72702 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","104 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992."]},{"key":"dc:title","label":"Title","values":["Real Even Symmetric Forms"]}]}],"canonical_facts":{"dc:contributor":["Reznick, B.,"],"dc:creator":["Harris, William Richard"],"dc:date":["2014-12-17T23:17:45Z","10000-01-01","1992"],"dc:description":["Let F$\\sb{n,m}$ denote the set of all real forms of degree m in n variables. In 1888, Hilbert proved that a form P $\\in$ F$\\sb{n,m}$ which is positive semidefinite (psd) must have a representation as a sum of squares (sos) of forms if and only if n = 2, m = 2, or (n,m) = (3,4). No concrete example of a psd form which is not sos was known until the late 1960's. We denote by S$\\sbsp{n,m}{e}$ the set of all real symmetric forms of degree m = 2d. Let PS$\\sbsp{n,m}{e}$ and $\\Sigma$S$\\sbsp{n,m}{e}$ denote the cones of psd and sos elements of S$\\sbsp{n,m}{e},$ respectively. For m = 2 or 4, these cones coincide. For m = 6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam and B. Reznick, Math. Z. 195 (1987), pp. 559-580.","We present an easily-checked, necessary and sufficient condition for an even symmetric n-ary octic to be in PS$\\sbsp{n,8}{e}$ and for an even symmetric ternary decic to be in PS$\\sbsp{3,10}{e},$ and also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10. We proceed to discuss the extremal elements of the cones PS$\\sbsp{3,8}{e},$ PS$\\sbsp{3,10}{e}$ and PS$\\sbsp{4,8}{e}.$ This leads to the question: how many of these extremal forms have sos representations? We prove that PS$\\sbsp{3,8}{e}$ = $\\Sigma$S$\\sbsp{3,8}{e},$ a companion result to Hilbert's theorem noted above, with regard to psd ternary quartics. We also demonstrate that neither PS$\\sbsp{3,10}{e}\\\\\\Sigma$S$\\sbsp{3,10}{e}$ nor PS$\\sbsp{4,8}{e}\\\\\\Sigma$S$\\sbsp{4,8}{e}$ is empty, providing many new examples of psd forms which are not sos.","We give a graphic representation with examples of ternary forms which also indicates whether or not an element of S$\\sbsp{3,8}{e}$ or S$\\sbsp{3,10}{e}$ is psd. We interpret elements of PS$\\sbsp{n,m}{e}$ as inequalities; in particular, we give all symmetric polynomial inequalities of degree $\\le$5 satisfied by the sides of a triangle.","Made available in DSpace on 2014-12-17T23:17:45Z (GMT). No. of bitstreams: 1 9305548.pdf: 3598174 bytes, checksum: afd93a2944786588bcf3375d5029a0a6 (MD5) Previous issue date: 1992","Embargo set by: Seth Robbins for item 72702 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","104 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992."],"dc:identifier":["http://hdl.handle.net/2142/72534","(UMI)AAI9305548"],"dc:subject":["Mathematics"],"dc:title":["Real Even Symmetric Forms"],"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"}