As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the multiplicity.<\/p>\n
Furthermore, this paper elaborates and proves some fundamental properties of the multiplicity, including local finiteness, consistency, perturbation invariance, and depth-deflatability.<\/p>\n
As a justification of this formulation, the multiplicity is proved to be consistent with the multiplicity defined in algebraic geometry for the special case of polynomial systems.<\/p>\n
The proposed algorithms can accurately compute the multiplicity and the multiple zeros using floating point arithmetic even if the nonlinear system is perturbed.<\/p>","DOI":"10.1090\/s0025-5718-2011-02462-2","type":"journal-article","created":{"date-parts":[[2011,2,3]],"date-time":"2011-02-03T12:50:37Z","timestamp":1296737437000},"page":"2143-2168","source":"Crossref","is-referenced-by-count":54,"title":["Multiple zeros of nonlinear systems"],"prefix":"10.1090","volume":"80","author":[{"given":"Barry","family":"Dayton","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tien-Yien","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhonggang","family":"Zeng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,2,3]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1016\/j.jco.2006.04.003","article-title":"A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set","volume":"22","author":"Bates, Dan","year":"2006","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"5","key":"2","doi-asserted-by":"publisher","first-page":"3608","DOI":"10.1137\/08073264X","article-title":"A numerical local dimensions test for points on the solution set of a system of polynomial equations","volume":"47","author":"Bates, Daniel J.","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"3","series-title":"Grundlehren der Mathematischen Wissenschaften","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-8159-4","volume-title":"Methods of bifurcation theory","volume":"251","author":"Chow, Shui Nee","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906649"},{"key":"4","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Using algebraic geometry","volume":"185","author":"Cox, David A.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/0387207066","edition":"2"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1145\/1073884.1073902","article-title":"Computing the multiplicity structure in solving polynomial systems","author":"Dayton, Barry H.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/1595930957"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971446","volume-title":"Applied numerical linear algebra","author":"Demmel, James W.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713897"},{"issue":"4","key":"7","doi-asserted-by":"crossref","first-page":"399","DOI":"10.24033\/bsmf.1879","article-title":"G\u00e9om\u00e9trie des points \u00e9pais","volume":"106","author":"Emsalem, Jacques","year":"1978","journal-title":"Bull. Soc. Math. France","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"key":"8","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02421-8","volume-title":"Intersection theory","volume":"2","author":"Fulton, William","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/3540121765"},{"key":"9","series-title":"Johns Hopkins Studies in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","author":"Golub, Gene H.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/080185413X","edition":"3"},{"key":"10","series-title":"B. I. Hochschultaschenb\\\"{u}cher","volume-title":"Algebraische Geometrie. $2$. Teil: Arithmetische Theorie der Polynomringe","volume":"737\/737","author":"Gr\u00f6bner, Wolfgang","year":"1970"},{"key":"11","isbn-type":"print","volume-title":"A {\\bf Singular} introduction to commutative algebra","author":"Greuel, Gert-Martin","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540735410","edition":"2"},{"key":"12","unstructured":"G.-M. Greuel, G. Pfister, and H. Sch\u00f6nemann, Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, Univ. of Kaiserslautern, 2005."},{"issue":"221","key":"13","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1090\/S0025-5718-98-00906-5","article-title":"Numerical calculation of the multiplicity of a solution to algebraic equations","volume":"67","author":"Kobayashi, Hidetsune","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","isbn-type":"print","volume-title":"Computational commutative algebra. 2","author":"Kreuzer, Martin","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540255277"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1016\/j.jmaa.2007.05.049","article-title":"Determining dimension of the solution component that contains a computed zero of a polynomial system","volume":"338","author":"Kuo, Y. C.","year":"2008","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/BF01447495","article-title":"Zur Theorie der moduln und Ideale","volume":"60","author":"Lasker, E.","year":"1905","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"1-3","key":"17","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/j.tcs.2006.02.018","article-title":"Newton\u2019s method with deflation for isolated singularities of polynomial systems","volume":"359","author":"Leykin, Anton","year":"2006","journal-title":"Theoret. Comput. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0304-3975","issn-type":"print"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/978-0-387-75155-9_5","article-title":"Higher-order deflation for polynomial systems with isolated singular solutions","author":"Leykin, Anton","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387751542"},{"issue":"2","key":"19","first-page":"97","article-title":"A method to solve algebraic equations up to multiplicities via Ritt-Wu\u2019s characteristic sets","volume":"5","author":"Li, Bang-He","year":"2003","journal-title":"Acta Anal. Funct. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1009-1327","issn-type":"print"},{"issue":"4","key":"20","doi-asserted-by":"publisher","first-page":"918","DOI":"10.1137\/S0895479803435282","article-title":"A rank-revealing method with updating, downdating, and applications","volume":"26","author":"Li, T. Y.","year":"2005","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"21","series-title":"Cambridge Mathematical Library","isbn-type":"print","volume-title":"The algebraic theory of modular systems","author":"Macaulay, F. S.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0521455626"},{"issue":"8","key":"22","doi-asserted-by":"publisher","first-page":"3283","DOI":"10.1090\/S0002-9947-96-01671-6","article-title":"On multiplicities in polynomial system solving","volume":"348","author":"Marinari, M. G.","year":"1996","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"23","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107340954","volume-title":"Solving polynomial equation systems. II","volume":"99","author":"Mora, Teo","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521811569"},{"key":"24","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1016\/S0022-4049(97)00023-6","article-title":"Isolated points, duality and residues","volume":"117\/118","author":"Mourrain, B.","year":"1997","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/0022-247X(87)90304-0","article-title":"Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations","volume":"123","author":"Ojika, Takeo","year":"1987","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"3","key":"26","doi-asserted-by":"publisher","first-page":"572","DOI":"10.1006\/jcom.2000.0554","article-title":"Numerical homotopies to compute generic points on positive dimensional algebraic sets","volume":"16","author":"Sommese, Andrew J.","year":"2000","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/0001-8708(78)90045-2","article-title":"Hilbert functions of graded algebras","volume":"28","author":"Stanley, Richard P.","year":"1978","journal-title":"Advances in Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"key":"28","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1145\/281508.281525","article-title":"Singular systems of polynomials","author":"Stetter, Hans J.","year":"1998"},{"key":"29","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717976","volume-title":"Numerical polynomial algebra","author":"Stetter, Hans J.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0898715571"},{"key":"30","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/046","volume-title":"Several complex variables with connections to algebraic geometry and Lie groups","volume":"46","author":"Taylor, Joseph L.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/082183178X"},{"key":"31","unstructured":"G. H. Thallinger, Analysis of Zero Clusters in Multivariate Polynomial Systems, Diploma Thesis, Tech. Univ. Vienna, 1996."},{"key":"32","isbn-type":"print","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1145\/1390768.1390812","article-title":"Computing the multiplicity structure from geometric involutive form","author":"Wu, Xiaoli","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9781595939043"},{"key":"33","series-title":"The University Series in Higher Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-29244-0","volume-title":"Commutative algebra. Vol. II","author":"Zariski, Oscar","year":"1960"},{"issue":"250","key":"34","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1090\/S0025-5718-04-01692-8","article-title":"Computing multiple roots of inexact polynomials","volume":"74","author":"Zeng, Zhonggang","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"35","doi-asserted-by":"crossref","unstructured":"\\bysame, ApaTools: A Maple and Matlab toolbox for approximate polynomial algebra, in Software for Algebraic Geometry, IMA Volume 148, M. Stillman, N. Takayama, and J. Verschelde, eds., Springer, 2008, pp.149\u2013167.","DOI":"10.1007\/978-0-387-78133-4_9"},{"key":"36","unstructured":"\\bysame, The closedness subspace method for computing the multiplicity structure of a polynomial system. to appear: Interactions between Classical and Numerical Algebraic Geometry, Contemporary Mathematics series, American Mathematical Society, 2009."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02462-2\/S0025-5718-2011-02462-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02462-2\/S0025-5718-2011-02462-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:57:31Z","timestamp":1776790651000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02462-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,3]]},"references-count":36,"journal-issue":{"issue":"276","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["S0025-5718-2011-02462-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02462-2","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":[[2011,2,3]]}}}