{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71216"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71216","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some Extensions of the Skolem-Mahler-Lech Theorem","abstract":"The Skolem-Mahler-Lech theorem states that if the Taylor series expansion (about the origin) of a rational function has infinitely many zero coefficients, then the set of indices of these zero coefficients forms a finite union of arithmetic progressions modulo a finite set. Since rational functions satisfy linear differential equations of order 0 with polynomial coefficients, it is natural to conjecture that the Skolem-Mahler-Lech theorem also holds for functions satisfying linear differential equations with polynomial coefficients of higher orders. In this thesis, we treat the case of first order differential equations. We are able to answer affirmatively parts of this conjecture, for example, we are able to show that functions satisfying first order linear homogeneous differential equations with polynomial coefficients of Fuchsian type are skomal. We also extend the Skolem-Mahler-Lech theorem in some other directions from the case of rational functions, and obtain certain arithmetical properties for functions satisfying linear differential equations with polynomial coefficients.","abstract_html":"The Skolem-Mahler-Lech theorem states that if the Taylor series expansion (about the origin) of a rational function has infinitely many zero coefficients, then the set of indices of these zero coefficients forms a finite union of arithmetic progressions modulo a finite set. Since rational functions satisfy linear differential equations of order 0 with polynomial coefficients, it is natural to conjecture that the Skolem-Mahler-Lech theorem also holds for functions satisfying linear differential equations with polynomial coefficients of higher orders. In this thesis, we treat the case of first order differential equations. We are able to answer affirmatively parts of this conjecture, for example, we are able to show that functions satisfying first order linear homogeneous differential equations with polynomial coefficients of Fuchsian type are skomal. We also extend the Skolem-Mahler-Lech theorem in some other directions from the case of rational functions, and obtain certain arithmetical properties for functions satisfying linear differential equations with polynomial coefficients.","abstract_has_math":false,"creators":["Laohakosol, Vichian"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:08Z","date_published":"2014-12-16T06:18:08Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8409980"],"render_values":[{"text":"(UMI)AAI8409980","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71216","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Laohakosol, Vichian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:08Z","10000-01-01","1983"]},{"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/2142/71216","(UMI)AAI8409980"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The Skolem-Mahler-Lech theorem states that if the Taylor series expansion (about the origin) of a rational function has infinitely many zero coefficients, then the set of indices of these zero coefficients forms a finite union of arithmetic progressions modulo a finite set. Since rational functions satisfy linear differential equations of order 0 with polynomial coefficients, it is natural to conjecture that the Skolem-Mahler-Lech theorem also holds for functions satisfying linear differential equations with polynomial coefficients of higher orders. In this thesis, we treat the case of first order differential equations. We are able to answer affirmatively parts of this conjecture, for example, we are able to show that functions satisfying first order linear homogeneous differential equations with polynomial coefficients of Fuchsian type are skomal. We also extend the Skolem-Mahler-Lech theorem in some other directions from the case of rational functions, and obtain certain arithmetical properties for functions satisfying linear differential equations with polynomial coefficients.","Made available in DSpace on 2014-12-16T06:18:08Z (GMT). No. of bitstreams: 1 8409980.pdf: 2448761 bytes, checksum: 322630d7cf92b7e1d1f07a962508f92e (MD5) Previous issue date: 1983","Embargo set by: Seth Robbins for item 71382 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","97 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983."]},{"key":"dc:title","label":"Title","values":["Some Extensions of the Skolem-Mahler-Lech Theorem"]}]}],"canonical_facts":{"dc:creator":["Laohakosol, Vichian"],"dc:date":["2014-12-16T06:18:08Z","10000-01-01","1983"],"dc:description":["The Skolem-Mahler-Lech theorem states that if the Taylor series expansion (about the origin) of a rational function has infinitely many zero coefficients, then the set of indices of these zero coefficients forms a finite union of arithmetic progressions modulo a finite set. Since rational functions satisfy linear differential equations of order 0 with polynomial coefficients, it is natural to conjecture that the Skolem-Mahler-Lech theorem also holds for functions satisfying linear differential equations with polynomial coefficients of higher orders. In this thesis, we treat the case of first order differential equations. We are able to answer affirmatively parts of this conjecture, for example, we are able to show that functions satisfying first order linear homogeneous differential equations with polynomial coefficients of Fuchsian type are skomal. We also extend the Skolem-Mahler-Lech theorem in some other directions from the case of rational functions, and obtain certain arithmetical properties for functions satisfying linear differential equations with polynomial coefficients.","Made available in DSpace on 2014-12-16T06:18:08Z (GMT). No. of bitstreams: 1 8409980.pdf: 2448761 bytes, checksum: 322630d7cf92b7e1d1f07a962508f92e (MD5) Previous issue date: 1983","Embargo set by: Seth Robbins for item 71382 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","97 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983."],"dc:identifier":["http://hdl.handle.net/2142/71216","(UMI)AAI8409980"],"dc:subject":["Mathematics"],"dc:title":["Some Extensions of the Skolem-Mahler-Lech Theorem"],"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":"2026-07-22T22:26:04Z"}