{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1927"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1927","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"Gröbner Bases and Syzygy Modules","abstract":"The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger's algorithm, and syzygy modules. Specially, we present several equivalent definitions for Gröbner bases and prove how to compute a Gröbner basis for an ideal I of A = k[x1, x2, • • • , xn] generated by {fl, f2, • • • , f8} through Buchberger's algorithm. As an application of Gröbner bases, we present a standard method (see [1]) to compute the syzygy module of a set {f1, f2, • • • , f8} of polynomials, illustrated with original examples. Finally, we implement these examples on the computer using the Mathematica package of [4].","abstract_html":"The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger's algorithm, and syzygy modules. Specially, we present several equivalent definitions for Gröbner bases and prove how to compute a Gröbner basis for an ideal I of A = k[x1, x2, • • • , xn] generated by {fl, f2, • • • , f8} through Buchberger's algorithm. As an application of Gröbner bases, we present a standard method (see [1]) to compute the syzygy module of a set {f1, f2, • • • , f8} of polynomials, illustrated with original examples. Finally, we implement these examples on the computer using the Mathematica package of [4].","abstract_has_math":false,"creators":["Zhao, Yonggan"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1995,"date_issued":"1995-08-01T07:00:00Z","date_published":"1995-08-01T07:00:00Z","updated_at":"2026-07-24T06:07:54Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/924","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Zhao, Yonggan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"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":["https://digitalcommons.wku.edu/theses/924"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger's algorithm, and syzygy modules. Specially, we present several equivalent definitions for Gröbner bases and prove how to compute a Gröbner basis for an ideal I of A = k[x1, x2, • • • , xn] generated by {fl, f2, • • • , f8} through Buchberger's algorithm. As an application of Gröbner bases, we present a standard method (see [1]) to compute the syzygy module of a set {f1, f2, • • • , f8} of polynomials, illustrated with original examples. Finally, we implement these examples on the computer using the Mathematica package of [4]."]},{"key":"dc:title","label":"Title","values":["Gröbner Bases and Syzygy Modules"]}]}],"canonical_facts":{"dc:creator":["Zhao, Yonggan"],"dc:description.abstract":["The theory of Gröbner bases has become a useful tool in computational commutative algebra. In this paper, we outline some basic results, as they are found in [1], including the concepts of terms ordering, multivariable polynomial division, Gröbner bases, Buchberger's algorithm, and syzygy modules. Specially, we present several equivalent definitions for Gröbner bases and prove how to compute a Gröbner basis for an ideal I of A = k[x1, x2, • • • , xn] generated by {fl, f2, • • • , f8} through Buchberger's algorithm. As an application of Gröbner bases, we present a standard method (see [1]) to compute the syzygy module of a set {f1, f2, • • • , f8} of polynomials, illustrated with original examples. Finally, we implement these examples on the computer using the Mathematica package of [4]."],"dc:identifier":["https://digitalcommons.wku.edu/theses/924"],"dc:subject":["Mathematics"],"dc:title":["Gröbner Bases and Syzygy Modules"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:54Z"}