{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T02:04:49Z","timestamp":1760061889166,"version":"3.40.3"},"publisher-location":"Cham","reference-count":46,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319663197"},{"type":"electronic","value":"9783319663203"}],"license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017]]},"DOI":"10.1007\/978-3-319-66320-3_28","type":"book-chapter","created":{"date-parts":[[2017,8,29]],"date-time":"2017-08-29T07:34:27Z","timestamp":1503992067000},"page":"392-405","source":"Crossref","is-referenced-by-count":0,"title":["Computing Multiple Zeros of Polynomial Systems: Case of Breadth One (Invited Talk)"],"prefix":"10.1007","author":[{"given":"Lihong","family":"Zhi","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,8,30]]},"reference":[{"key":"28_CR1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0701-6","volume-title":"Complexity and Real Computation","author":"L Blum","year":"1998","unstructured":"Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998)"},{"issue":"2","key":"28_CR2","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1137\/S1052623493246288","volume":"7","author":"X Chen","year":"1997","unstructured":"Chen, X., Nashed, Z., Qi, L.: Convergence of Newton\u2019s method for singular smooth and nonsmooth equations using adaptive outer inverses. SIAM J. Optim. 7(2), 445\u2013462 (1997)","journal-title":"SIAM J. Optim."},{"key":"28_CR3","doi-asserted-by":"crossref","unstructured":"Corless, R.M., Gianni, P.M., Trager, B.M.: A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In: K\u00fcchlin, W.W. (ed) Proceedings of ISSAC 1997, pp. 133\u2013140. ACM, New York (1997)","DOI":"10.1145\/258726.258767"},{"key":"28_CR4","doi-asserted-by":"crossref","first-page":"2143","DOI":"10.1090\/S0025-5718-2011-02462-2","volume":"80","author":"B Dayton","year":"2011","unstructured":"Dayton, B., Li, T., Zeng, Z.: Multiple zeros of nonlinear systems. Math. Comput. 80, 2143\u20132168 (2011)","journal-title":"Math. Comput."},{"key":"28_CR5","doi-asserted-by":"crossref","unstructured":"Dayton, B., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Kauers, M. (ed) Proceedings of ISSAC 2005, pp. 116\u2013123. ACM, New York (2005)","DOI":"10.1145\/1073884.1073902"},{"key":"28_CR6","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1137\/0717009","volume":"17","author":"DW Decker","year":"1980","unstructured":"Decker, D.W., Kelley, C.T.: Newton\u2019s method at singular points I. SIAM J. Numer. Anal. 17, 66\u201370 (1980)","journal-title":"SIAM J. Numer. Anal."},{"key":"28_CR7","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1137\/0717039","volume":"17","author":"DW Decker","year":"1980","unstructured":"Decker, D.W., Kelley, C.T.: Newton\u2019s method at singular points II. SIAM J. Numer. Anal. 17, 465\u2013471 (1980)","journal-title":"SIAM J. Numer. Anal."},{"key":"28_CR8","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1137\/0719012","volume":"19","author":"DW Decker","year":"1982","unstructured":"Decker, D.W., Kelley, C.T.: Convergence acceleration for Newton\u2019s method at singular points. SIAM J. Numer. Anal. 19, 219\u2013229 (1982)","journal-title":"SIAM J. Numer. Anal."},{"issue":"233","key":"28_CR9","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1090\/S0025-5718-00-01194-7","volume":"70","author":"JP Dedieu","year":"2001","unstructured":"Dedieu, J.P., Shub, M.: On simple double zeros and badly conditioned zeros of analytic functions of n variables. Math. Comput. 70(233), 319\u2013327 (2001)","journal-title":"Math. Comput."},{"issue":"3","key":"28_CR10","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s10208-004-0144-z","volume":"5","author":"M Giusti","year":"2005","unstructured":"Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.C.: On location and approximation of clusters of zeros of analytic functions. Found. Comput. Math. 5(3), 257\u2013311 (2005)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"28_CR11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10208-004-0159-5","volume":"7","author":"M Giusti","year":"2007","unstructured":"Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.C.: On location and approximation of clusters of zeros: case of embedding dimension one. Found. Comput. Math. 7(1), 1\u201358 (2007)","journal-title":"Found. Comput. Math."},{"issue":"4","key":"28_CR12","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1137\/1027141","volume":"27","author":"A Griewank","year":"1985","unstructured":"Griewank, A.: On solving nonlinear equations with simple singularities or nearly singular solutions. SIAM Rev. 27(4), 537\u2013563 (1985)","journal-title":"SIAM Rev."},{"key":"28_CR13","unstructured":"Griewank, A.: Analysis and modification of Newton\u2019s method at singularities. Australian National University, thesis (1980)"},{"key":"28_CR14","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1137\/0718011","volume":"18","author":"A Griewank","year":"1981","unstructured":"Griewank, A., Osborne, M.R.: Newton\u2019s method for singular problems when the dimension of the null space is \n            $$>1$$\n          . SIAM J. Numer. Anal. 18, 145\u2013149 (1981)","journal-title":"SIAM J. Numer. Anal."},{"key":"28_CR15","unstructured":"Hao, Z., Jiang, W., Li, N., Zhi, L.: Computing simple multiple zeros of polynomial systems (2017). \nhttps:\/\/www.arxiv.org\/pdf\/1703.03981.pdf"},{"issue":"4","key":"28_CR16","doi-asserted-by":"crossref","first-page":"28:1","DOI":"10.1145\/2331130.2331136","volume":"38","author":"JD Hauenstein","year":"2012","unstructured":"Hauenstein, J.D., Sottile, F.: Algorithm 921: AlphaCertified: certifying solutions to polynomial systems. ACM Trans. Math. Softw. 38(4), 28:1\u201328:20 (2012)","journal-title":"ACM Trans. Math. Softw."},{"issue":"3","key":"28_CR17","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF02234767","volume":"4","author":"R Krawczyk","year":"1969","unstructured":"Krawczyk, R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4(3), 187\u2013201 (1969)","journal-title":"Computing"},{"issue":"3","key":"28_CR18","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s102080010026","volume":"2","author":"G Lecerf","year":"2002","unstructured":"Lecerf, G.: Quadratic Newton iteration for systems with multiplicity. Found. Comput. Math. 2(3), 247\u2013293 (2002)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"28_CR19","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.tcs.2006.02.018","volume":"359","author":"A Leykin","year":"2006","unstructured":"Leykin, A., Verschelde, J., Zhao, A.: Newton\u2019s method with deflation for isolated singularities of polynomial systems. Theoret. Comput. Sci. 359(1), 111\u2013122 (2006)","journal-title":"Theoret. Comput. Sci."},{"key":"28_CR20","series-title":"The IMA Volumes in Mathematics and its Applications","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/978-0-387-75155-9_5","volume-title":"Algorithms in Algebraic Geometry","author":"A Leykin","year":"2008","unstructured":"Leykin, A., Verschelde, J., Zhao, A.: Higher-order deflation for polynomial systems with isolated singular solutions. In: Dickenstein, A., Schreyer, F.O., Sommese, A.J. (eds.) Algorithms in Algebraic Geometry. IMA, vol. 146, pp. 79\u201397. Springer, New York (2008)"},{"key":"28_CR21","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.jsc.2011.12.027","volume":"47","author":"N Li","year":"2012","unstructured":"Li, N., Zhi, L.: Compute the multiplicity structure of an isolated singular solution: case of breadth one. J. Symb. Comput. 47, 700\u2013710 (2012)","journal-title":"J. Symb. Comput."},{"issue":"1","key":"28_CR22","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1137\/110827247","volume":"50","author":"N Li","year":"2012","unstructured":"Li, N., Zhi, L.: Computing isolated singular solutions of polynomial systems: case of breadth one. SIAM J. Numer. Anal. 50(1), 354\u2013372 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"28_CR23","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.tcs.2012.10.028","volume":"479","author":"N Li","year":"2013","unstructured":"Li, N., Zhi, L.: Verified error bounds for isolated singular solutions of polynomial systems: case of breadth one. Theoret. Comput. Sci. 479, 163\u2013173 (2013)","journal-title":"Theoret. Comput. Sci."},{"issue":"4","key":"28_CR24","doi-asserted-by":"crossref","first-page":"1623","DOI":"10.1137\/120902914","volume":"52","author":"N Li","year":"2014","unstructured":"Li, N., Zhi, L.: Verified error bounds for isolated singular solutions of polynomial systems. SIAM J. Numer. Anal. 52(4), 1623\u20131640 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"28_CR25","doi-asserted-by":"crossref","unstructured":"Mantzaflaris, A., Mourrain, B.: Deflation and certified isolation of singular zeros of polynomial systems. In: Leykin, A. (ed.) Proceedings of ISSAC 2011, pp. 249\u2013256. ACM, New York (2011)","DOI":"10.1145\/1993886.1993925"},{"key":"28_CR26","doi-asserted-by":"crossref","unstructured":"Marinari, M.G., Mora, T., M\u00f6ller, H.M.: Gr\u00f6bner duality and multiplicities in polynomial system solving. In: Proceedings of ISSAC 1995, pp. 167\u2013179. ACM, New York (1995)","DOI":"10.1145\/220346.220368"},{"issue":"4","key":"28_CR27","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1137\/0714040","volume":"14","author":"RE Moore","year":"1977","unstructured":"Moore, R.E.: A test for existence of solutions to nonlinear systems. SIAM J. Numer. Anal. 14(4), 611\u2013615 (1977)","journal-title":"SIAM J. Numer. Anal."},{"key":"28_CR28","first-page":"469","volume":"117","author":"B Mourrain","year":"1996","unstructured":"Mourrain, B.: Isolated points, duality and residues. J. Pure Appl. Algebra 117, 469\u2013493 (1996). 117","journal-title":"J. Pure Appl. Algebra"},{"issue":"1","key":"28_CR29","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0022-247X(87)90304-0","volume":"123","author":"T Ojika","year":"1987","unstructured":"Ojika, T.: Modified deflation algorithm for the solution of singular problems. i. a system of nonlinear algebraic equations. J. Math. Anal. Appl. 123(1), 199\u2013221 (1987)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"28_CR30","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1016\/0022-247X(83)90055-0","volume":"96","author":"T Ojika","year":"1983","unstructured":"Ojika, T., Watanabe, S., Mitsui, T.: Deflation algorithm for the multiple roots of a system of nonlinear equations. J. Math. Anal. Appl. 96(2), 463\u2013479 (1983)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"28_CR31","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/BF02165226","volume":"9","author":"LB Rall","year":"1966","unstructured":"Rall, L.B.: Convergence of the Newton process to multiple solutions. Numer. Math. 9(1), 23\u201337 (1966)","journal-title":"Numer. Math."},{"issue":"5","key":"28_CR32","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1137\/0715064","volume":"15","author":"GW Reddien","year":"1978","unstructured":"Reddien, G.W.: On Newton\u2019s method for singular problems. SIAM J. Numer. Anal. 15(5), 993\u2013996 (1978)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"28_CR33","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/0898-1221(79)90061-0","volume":"5","author":"GW Reddien","year":"1979","unstructured":"Reddien, G.W.: Newton\u2019s method and high order singularities. Comput. Math. Appl. 5(2), 79\u201386 (1979)","journal-title":"Comput. Math. Appl."},{"key":"28_CR34","doi-asserted-by":"crossref","unstructured":"Rump, S.M.: Solving algebraic problems with high accuracy. In: Proceedings of the Symposium on A New Approach to Scientific Computation, pp. 51\u2013120. Academic Press Professional Inc., San Diego (1983)","DOI":"10.1016\/B978-0-12-428660-3.50010-0"},{"issue":"3","key":"28_CR35","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s11075-009-9339-3","volume":"54","author":"SM Rump","year":"2010","unstructured":"Rump, S.M., Graillat, S.: Verified error bounds for multiple roots of systems of nonlinear equations. Numer. Algorithms 54(3), 359\u2013377 (2010)","journal-title":"Numer. Algorithms"},{"issue":"2","key":"28_CR36","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.apnum.2004.09.029","volume":"54","author":"YQ Shen","year":"2005","unstructured":"Shen, Y.Q., Ypma, T.J.: Newton\u2019s method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian. Appl. Numer. Math. 54(2), 256\u2013265 (2005)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"28_CR37","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1137\/0733008","volume":"33","author":"M Shub","year":"1996","unstructured":"Shub, M., Smale, S.: Complexity of bezout\u2019s theorem IV: probability of success; extensions. SIAM J. Numer. Anal. 33(1), 128\u2013148 (1996)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"28_CR38","doi-asserted-by":"crossref","first-page":"107","DOI":"10.24033\/asens.1486","volume":"18","author":"M Shub","year":"1985","unstructured":"Shub, M., Smale, S.: Computational complexity: on the geometry of polynomials and a theory of cost: I. Ann. Sci. \u00c9c. Norm. Sup\u00e9r. 18(1), 107\u2013142 (1985)","journal-title":"Ann. Sci. \u00c9c. Norm. Sup\u00e9r."},{"issue":"1","key":"28_CR39","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1137\/0215011","volume":"15","author":"M Shub","year":"1986","unstructured":"Shub, M., Smale, S.: Computational complexity: on the geometry of polynomials and a theory of cost: II. SIAM J. Comput. 15(1), 145\u2013161 (1986)","journal-title":"SIAM J. Comput."},{"issue":"1","key":"28_CR40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0273-0979-1981-14858-8","volume":"4","author":"S Smale","year":"1981","unstructured":"Smale, S.: The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. 4(1), 1\u201336 (1981)","journal-title":"Bull. Amer. Math. Soc."},{"key":"28_CR41","volume-title":"The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics","author":"S Smale","year":"1986","unstructured":"Smale, S.: Newton\u2019s method estimates from data at one point. In: Ewing, R.E., Gross, K.I., Martin, C.F. (eds.) The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics. Springer, New York (1986)"},{"key":"28_CR42","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898717976","volume-title":"Numerical Polynomial Algebra","author":"H Stetter","year":"2004","unstructured":"Stetter, H.: Numerical Polynomial Algebra. SIAM, Philadelphia (2004)"},{"issue":"2","key":"28_CR43","first-page":"135","volume":"33","author":"X Wang","year":"1990","unstructured":"Wang, X., Han, D.: On dominating sequence method in the point estimate and smale theorem. Sci. China Ser. A 33(2), 135\u2013144 (1990)","journal-title":"Sci. China Ser. A"},{"key":"28_CR44","doi-asserted-by":"crossref","unstructured":"Wu, X., Zhi, L.: Computing the multiplicity structure from geometric involutive form. In: Jeffrey, D. (ed) Proceedings of ISSAC 2008, pp. 325\u2013332. ACM, New York (2008)","DOI":"10.1145\/1390768.1390812"},{"issue":"3","key":"28_CR45","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.jsc.2011.10.001","volume":"47","author":"X Wu","year":"2012","unstructured":"Wu, X., Zhi, L.: Determining singular solutions of polynomial systems via symbolic-numeric reduction to geometric involutive forms. J. Symb. Comput. 47(3), 227\u2013238 (2012)","journal-title":"J. Symb. Comput."},{"issue":"1","key":"28_CR46","first-page":"16","volume":"7","author":"N Yamamoto","year":"1984","unstructured":"Yamamoto, N.: Regularization of solutions of nonlinear equations with singular Jacobian matrices. J. Inf. Process. 7(1), 16\u201321 (1984)","journal-title":"J. Inf. Process."}],"container-title":["Lecture Notes in Computer Science","Computer Algebra in Scientific Computing"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-66320-3_28","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,8,30]],"date-time":"2017-08-30T03:31:17Z","timestamp":1504063877000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-319-66320-3_28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319663197","9783319663203"],"references-count":46,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-66320-3_28","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2017]]}}}