{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:01:59Z","timestamp":1776830519522,"version":"3.51.2"},"reference-count":6,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,7,10]],"date-time":"2001-07-10T00:00:00Z","timestamp":994723200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. In their study of the complexity of B\u00e9zout\u2019s theorem, M. Shub and S. Smale prove that the condition number of a polynomial system is equal to the inverse of the distance from this polynomial system to the nearest ill-conditioned one. Here we explain how this result can be extended to underdetermined systems of polynomials (that is with less equations than unknowns).<\/p>","DOI":"10.1090\/s0025-5718-00-00934-0","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T17:02:26Z","timestamp":1121101346000},"page":"329-335","source":"Crossref","is-referenced-by-count":5,"title":["A condition number theorem for underdetermined polynomial systems"],"prefix":"10.1090","volume":"70","author":[{"given":"J\u00e9r\u00f4me","family":"D\u00e9got","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,7,10]]},"reference":[{"issue":"7","key":"1","doi-asserted-by":"publisher","first-page":"2607","DOI":"10.2307\/2154842","article-title":"Differential identities","volume":"347","author":"Beauzamy, Bernard","year":"1995","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/bf02562881","article-title":"Approximate solutions of numerical problems, condition number analysis and condition number theorem","author":"Dedieu, Jean-Pierre","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0821805304"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/BF01400115","article-title":"On condition numbers and the distance to the nearest ill-posed problem","volume":"51","author":"Demmel, James Weldon","year":"1987","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"C. Eckart, G. Young, The approximation of one matrix by another of lower rank, Psychometrika 1, (1936), pp 211-218.","DOI":"10.1007\/BF02288367"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.2307\/2159535","article-title":"An inequality for products of polynomials","volume":"117","author":"Reznick, Bruce","year":"1993","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"459","DOI":"10.2307\/2152805","article-title":"Complexity of B\u00e9zout\u2019s theorem. I. Geometric aspects","volume":"6","author":"Shub, Michael","year":"1993","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-00934-0\/S0025-5718-00-00934-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-00934-0\/S0025-5718-00-00934-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:30:46Z","timestamp":1776724246000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-233\/S0025-5718-00-00934-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,7,10]]},"references-count":6,"journal-issue":{"issue":"233","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["S0025-5718-00-00934-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-00934-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2000,7,10]]}}}