{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T05:27:09Z","timestamp":1775021229699,"version":"3.50.1"},"reference-count":6,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[1983,5,1]],"date-time":"1983-05-01T00:00:00Z","timestamp":420595200000},"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":[[1983,5]]},"abstract":"<jats:p>The problem of finding a modular algorithm for constructing Gr\u00f6bner-bases is of interest to many computer algebraists. In particular, given a prime p and a set of (multivariate) polynomials with integer coefficients, it has been queried if the number of basis polynomials in a minimal normed Gr\u00f6bner-basis for the polynomial ideal generated mod p has to be less than or equal to the corresponding number for the polynomial ideal generated over the rationals. In this paper we answer this question and related questions concerning the modular approach to Gr\u00f6bner-bases, illustrating with several interesting examples, and we propose a criterion for determining \"luckiness\" of primes in the binomial case.<\/jats:p>","DOI":"10.1145\/1089330.1089336","type":"journal-article","created":{"date-parts":[[2007,1,17]],"date-time":"2007-01-17T18:32:02Z","timestamp":1169058722000},"page":"28-32","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":18,"title":["Some comments on the modular approach to Gr\u00f6bner-bases"],"prefix":"10.1145","volume":"17","author":[{"given":"G. L.","family":"Ebert","sequence":"first","affiliation":[{"name":"University of Delaware, Newark, Delaware"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[1983,5]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/321662.321664"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/1088216.1088219"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1088222.1088224"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.5555\/646670.699008"},{"key":"e_1_2_1_5_1","volume-title":"Inst. f\u00fcr Mathematik, Univ. Linz","author":"Buchberger B.","year":"1981","unstructured":"B. Buchberger , \"H-bases and Gr\u00f6bner-bases for polynomial ideals\", Tech. Report Nr. CAMP 81-2.0 , Inst. f\u00fcr Mathematik, Univ. Linz ( 1981 ). B. Buchberger, \"H-bases and Gr\u00f6bner-bases for polynomial ideals\", Tech. Report Nr. CAMP 81-2.0, Inst. f\u00fcr Mathematik, Univ. Linz (1981)."},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/800205.806353"}],"container-title":["ACM SIGSAM Bulletin"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1089330.1089336","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1089330.1089336","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T16:08:23Z","timestamp":1750262903000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1089330.1089336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,5]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1983,5]]}},"alternative-id":["10.1145\/1089330.1089336"],"URL":"https:\/\/doi.org\/10.1145\/1089330.1089336","relation":{},"ISSN":["0163-5824"],"issn-type":[{"value":"0163-5824","type":"print"}],"subject":[],"published":{"date-parts":[[1983,5]]},"assertion":[{"value":"1983-05-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}