{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T17:09:09Z","timestamp":1745946549258,"version":"3.37.3"},"reference-count":7,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:00:00Z","timestamp":1665964800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:00:00Z","timestamp":1665964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100009044","name":"Technische Universit\u00e4t Kaiserslautern","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100009044","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math.Comput.Sci."],"published-print":{"date-parts":[[2022,12]]},"abstract":"Abstract<\/jats:title>We present new results on standard basis computations of a 0-dimensional ideal I<\/jats:italic> in a power series ring or in the localization of a polynomial ring over a computable field K<\/jats:italic>. We prove the semicontinuity of the \u201chighest corner\u201d in a family of ideals, parametrized by the spectrum of a Noetherian domain A<\/jats:italic>. This semicontinuity is used to design a new modular algorithm for computing a standard basis of I<\/jats:italic> if K<\/jats:italic> is the quotient field of A<\/jats:italic>. It uses the computation over the residue field of a \u201cgood\u201d prime ideal of A<\/jats:italic> to truncate high order terms in the subsequent computation over K<\/jats:italic>. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps $$A={\\mathbb {Z}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n =<\/mml:mo>\n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$A=k[t]$$<\/jats:tex-math>\n \n A<\/mml:mi>\n =<\/mml:mo>\n k<\/mml:mi>\n [<\/mml:mo>\n t<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, k<\/jats:italic> any field and t<\/jats:italic> a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for $$A={\\mathbb {Z}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n =<\/mml:mo>\n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular<\/jats:sc> and we present several examples illustrating its power.\n<\/jats:p>","DOI":"10.1007\/s11786-022-00539-2","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T06:02:20Z","timestamp":1665986540000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Using Semicontinuity for Standard Bases Computations"],"prefix":"10.1007","volume":"16","author":[{"given":"Gert-Martin","family":"Greuel","sequence":"first","affiliation":[]},{"given":"Gerhard","family":"Pfister","sequence":"additional","affiliation":[]},{"given":"Hans","family":"Sch\u00f6nemann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,17]]},"reference":[{"issue":"1","key":"539_CR1","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.jsc.2004.09.001","volume":"39","author":"J Abbott","year":"2005","unstructured":"Abbott, J., Kreuzer, M., Robbiano, L.: Computing Zero-dimensional Schemes. J. Symb. Comput. 39(1), 31\u201349 (2005)","journal-title":"J. Symb. Comput."},{"issue":"4","key":"539_CR2","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/S0747-7171(02)00140-2","volume":"35","author":"EA Arnold","year":"2003","unstructured":"Arnold, E.A.: Modular algorithms for computing Gr\u00f6bner bases. J. Symb. Comput. 35(4), 403\u2013419 (2003)","journal-title":"J. Symb. Comput."},{"key":"539_CR3","unstructured":"Decker, W., Greuel, G.-M., Pfister, G., Sch\u00f6nemann, H.: Singular 4-2-1 \u2013 A computer algebra system for polynomial computations (2021). http:\/\/www.singular.uni-kl.de"},{"key":"539_CR4","unstructured":"Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. With contributions by O. Bachmann, C. Lossen and H. Sch\u00f6nemann. 2nd Edition, Springer, Berlin (2008)"},{"key":"539_CR5","unstructured":"Greuel, G.-M., Pfister, G.: Semicontinuity of Singularity Invariants in Families of Formal Power Series, arXiv:1912.05263v3 (2020). To appear in the Proceedings of the N\u00e9methi60 Conference"},{"key":"539_CR6","volume-title":"Commutative Ring Theory","author":"H Matsumura","year":"1986","unstructured":"Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)"},{"key":"539_CR7","unstructured":"Pfister, G.: On modular computation of standard basis. An. Stiint. Univ. Ovidius Constan\u0163a, Ser. Mat. 15, No. 1, 129-138 (2007)"}],"container-title":["Mathematics in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11786-022-00539-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11786-022-00539-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11786-022-00539-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,19]],"date-time":"2023-09-19T06:22:49Z","timestamp":1695104569000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11786-022-00539-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,17]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["539"],"URL":"https:\/\/doi.org\/10.1007\/s11786-022-00539-2","relation":{},"ISSN":["1661-8270","1661-8289"],"issn-type":[{"type":"print","value":"1661-8270"},{"type":"electronic","value":"1661-8289"}],"subject":[],"published":{"date-parts":[[2022,10,17]]},"assertion":[{"value":"17 October 2022","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 September 2023","order":2,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":3,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Missing Open Access funding information has been added in the Funding Note.","order":4,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"21"}}