{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:42:03Z","timestamp":1750308123258,"version":"3.41.0"},"reference-count":8,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[1980,11,1]],"date-time":"1980-11-01T00:00:00Z","timestamp":341884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["SIGSAM Bull."],"published-print":{"date-parts":[[1980,11]]},"abstract":"<jats:p>In \/2\/ a certain type of bases (\"Gr\u00f6bner-Bases\") for polynomial ideals has been introduced whose usefulness stems from the fact that a number of important computability problems in the theory of polynomial ideals are reducible to the construction of bases of this type. The key to an algorithmic construction of Gr\u00f6bner-bases is a characterization theorem for Gr\u00f6bner-bases whose proof in \/2\/is rather complex.In this paper a simplified proof is given. The simplification is based on two new lemmas that are of some interest in themselves. The first lemma characterizes the congruence relation modulo a polynomial ideal as the reflexive-transitive closure of a particular reduction relation (\"M-reduction\") used in the definition of Gr\u00f6bner-bases and its inverse. The second lemma is a lemma on general reduction relations, which allows to guarantee the Church-Rosser property under very weak assumptions.<\/jats:p>","DOI":"10.1145\/1089235.1089238","type":"journal-article","created":{"date-parts":[[2007,1,17]],"date-time":"2007-01-17T18:32:02Z","timestamp":1169058722000},"page":"29-34","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":16,"title":["A simplified proof of the characterization theorem for Gr\u00f6bner-bases"],"prefix":"10.1145","volume":"14","author":[{"given":"L.","family":"Bachmair","sequence":"first","affiliation":[{"name":"Johannes Kepler Universit\u00e4t, Linz, Austria"}]},{"given":"B.","family":"Buchberger","sequence":"additional","affiliation":[{"name":"Johannes Kepler Universit\u00e4t, Linz, Austria"}]}],"member":"320","published-online":{"date-parts":[[1980,11]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","unstructured":"B. Buchberger Ph.D. Thesis Univ. Innsbruck 1965 (see also Aequationes mathematicae Vol. 4\/3 pp. 374--383 1970).  B. Buchberger Ph.D. Thesis Univ. Innsbruck 1965 (see also Aequationes mathematicae Vol. 4\/3 pp. 374--383 1970).","DOI":"10.1007\/BF01844169"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1088216.1088219"},{"key":"e_1_2_1_3_1","first-page":"3","article-title":"Reductions in the Construction of Gr\u00f6bner-Bases, in: Proc. EUROSAM 79, Marseille, (Lecture Notes","volume":"72","author":"Buchberger B.","year":"1979","journal-title":"Computer Science"},{"volume-title":"Institut f\u00fcr Mathematik","year":"1979","author":"Buchberger B.","key":"e_1_2_1_4_1"},{"volume-title":"18th IEEE Symp. on Found. Comp. Scie, 30--45","year":"1977","author":"Huet G.","key":"e_1_2_1_5_1"},{"key":"e_1_2_1_6_1","first-page":"263","volume-title":"Ed. J. Leech, Pergamon Press","author":"Knuth D.","year":"1970"},{"key":"e_1_2_1_7_1","doi-asserted-by":"crossref","unstructured":"C. Kollreider B. Buchberger An Improved Algorithmic Construction of Gr\u00f6bner-Bases for Polynomial Ideals Bericht Nr. 110 Institut f\u00fcr Mathematik Universit\u00e4t Linz 1978.  C. Kollreider B. Buchberger An Improved Algorithmic Construction of Gr\u00f6bner-Bases for Polynomial Ideals Bericht Nr. 110 Institut f\u00fcr Mathematik Universit\u00e4t Linz 1978.","DOI":"10.1145\/1088261.1088267"},{"key":"e_1_2_1_8_1","unstructured":"R. Loos personal communication 1979.  R. Loos personal communication 1979."}],"container-title":["ACM SIGSAM Bulletin"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1089235.1089238","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1089235.1089238","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T16:08:22Z","timestamp":1750262902000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1089235.1089238"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,11]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1980,11]]}},"alternative-id":["10.1145\/1089235.1089238"],"URL":"https:\/\/doi.org\/10.1145\/1089235.1089238","relation":{},"ISSN":["0163-5824"],"issn-type":[{"type":"print","value":"0163-5824"}],"subject":[],"published":{"date-parts":[[1980,11]]},"assertion":[{"value":"1980-11-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}