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Cell decomposition of almost smooth real algebraic surfaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-7186-1524
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2013 (English)In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 63, no 4, 645-678 p.Article in journal (Refereed) Published
Abstract [en]

Let Z be a two dimensional irreducible complex component of the solution set of a system of polynomial equations with real coefficients in N complex variables. This work presents a new numerical algorithm, based on homotopy continuation methods, that begins with a numerical witness set for Z and produces a decomposition into 2-cells of any almost smooth real algebraic surface contained in Z. Each 2-cell (a face) has a generic interior point and a boundary consisting of 1-cells (edges). Similarly, the 1-cells have a generic interior point and a vertex at each end. Each 1-cell and each 2-cell has an associated homotopy for moving the generic interior point to any other point in the interior of the cell, defining an invertible map from the parameter space of the homotopy to the cell. This work draws on previous results for the curve case. Once the cell decomposition is in hand, one can sample the 2-cells and 1-cells to any resolution, limited only by the computational resources available.

Place, publisher, year, edition, pages
2013. Vol. 63, no 4, 645-678 p.
Keyword [en]
Algebraic surface, Algebraic curve, Cell decomposition, Numerical algebraic geometry, Homotopy, Polynomial system
National Category
URN: urn:nbn:se:kth:diva-129124DOI: 10.1007/s11075-012-9646-yISI: 000323341900004ScopusID: 2-s2.0-84880506061OAI: diva2:650189
Swedish Research Council, NT:2010-5563

QC 20130920

Available from: 2013-09-20 Created: 2013-09-19 Last updated: 2013-09-20Bibliographically approved

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Di Rocco, Sandra
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