{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:35Z","timestamp":1776721955878,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>This paper contains a description of a program designed to find all the solutions of systems of two real equations in two real unknowns which uses detailed information about the critical set of the associated function from the plane to the plane. It turns out that the critical set and its image are highly structured, and this is employed in their numerical computation. The conceptual background and details of implementation are presented. The most important features of the program are the ability to provide global information about the function and the robustness derived from such topological information.<\/p>","DOI":"10.1090\/s0025-5718-96-00770-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1531-1552","source":"Crossref","is-referenced-by-count":10,"title":["The numerical inversion of functions from the plane to the plane"],"prefix":"10.1090","volume":"65","author":[{"given":"Iaci","family":"Malta","sequence":"first","affiliation":[]},{"given":"Nicolau","family":"Saldanha","sequence":"additional","affiliation":[]},{"given":"Carlos","family":"Tomei","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","series-title":"Lectures in Applied Mathematics","isbn-type":"print","volume-title":"Exploiting symmetry in applied and numerical analysis","volume":"29","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0821811347"},{"key":"2","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"Practical methods of optimization. Vol. 1","author":"Fletcher, R.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0471277118"},{"issue":"2","key":"3","first-page":"165","article-title":"Excellent maps with given folds and cusps","volume":"3","author":"Francis, George K.","year":"1977","journal-title":"Houston J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-1588","issn-type":"print"},{"key":"4","isbn-type":"print","volume-title":"Practical optimization","author":"Gill, Philip E.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0122839501"},{"key":"5","isbn-type":"print","first-page":"73","article-title":"Global homotopies and Newton methods","author":"Keller, Herbert B.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0122083601"},{"key":"6","series-title":"Springer Series in Computational Physics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-85957-1","volume-title":"Computational methods in bifurcation theory and dissipative structures","author":"Kub\u00ed\u010dek, M.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/038712070X"},{"key":"7","unstructured":"[7] I. Malta, N. C. Saldanha and C. Tomei, Critical sets of proper Whitney functions in the plane (to appear)"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/0022-0396(91)90107-K","article-title":"Singularities of vector fields arising from one-dimensional Riemann problems","volume":"94","author":"Malta, Iaci","year":"1991","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"9","volume-title":"Topology from the differentiable viewpoint","author":"Milnor, John W.","year":"1965"},{"key":"10","isbn-type":"print","volume-title":"Solving polynomial systems using continuation for engineering and scientific problems","author":"Morgan, Alexander","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0138223130"},{"key":"11","unstructured":"[11] V. Po\u00e9naru, Extending immersions of the circle (d\u2019apr\u00e8s Samuel Blank), Expos\u00e9 342, S\u00e9minaire Bourbaki 1967\u201368, Benjamin, NY, 1969."},{"key":"12","doi-asserted-by":"publisher","first-page":"307","DOI":"10.2307\/1997788","article-title":"A global theorem for singularities of maps between oriented 2-manifolds","volume":"236","author":"Quine, J. R.","year":"1978","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"13","unstructured":"[13] S. F. Troyer, Extending a boundary immersion to the disk with \ud835\udc5b holes, PhD Dissertation, Northeastern Univ., Boston, Mass., 1973"},{"key":"14","doi-asserted-by":"publisher","first-page":"23","DOI":"10.2307\/1989990","article-title":"Steinitz field towers for modular fields","volume":"46","author":"MacLane, Saunders","year":"1939","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00770-3\/S0025-5718-96-00770-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00770-3\/S0025-5718-96-00770-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:16:50Z","timestamp":1776719810000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00770-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":14,"journal-issue":{"issue":"216","published-print":{"date-parts":[[1996,10]]}},"alternative-id":["S0025-5718-96-00770-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00770-3","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":[[1996]]}}}