Cell decomposition of almost smooth real algebraic surfaces

Authors:
Gian Mario Besana;Sandra Rocco;Jonathan D. Hauenstein;Andrew J. Sommese;Charles W. Wampler
Affiliations:
College of Computing and Digital Media, DePaul University, Chicago, USA 60604;Department of Mathematics, KTH, Stockholm, Sweden 10044;Department of Mathematics, North Carolina State University, Raleigh, USA 27695;Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, USA 46556;General Motors Research and Development, Warren, USA 48090
Venue:
Numerical Algorithms
Year:
2013

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Abstract

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.