An efficient algorithm for the stratification and triangulation of an algebraic surface

Eric Berberich, Michael Kerber, Michael Sagraloff

Abstract: We present a method to compute the exact topology of a real algebraic surface S, implicitly given by a polynomial f∈Q[x,y,z]<math><mi is="true">f</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">Q</mi><mo stretchy="false" is="true">[</mo><mi is="true">x</mi><mo is="true">,</mo><mi is="true">y</mi><mo is="true">,</mo><mi is="true">z</mi><mo stretchy="false" is="true">]</mo></math> of arbitrary total degree N. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of S including critical points. We compute a stratification ΩS<math><msub is="true"><mi is="true">Ω</mi><mi is="true">S</mi></msub></math> of S into O(N5)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mi is="true">N</mi><mn is="true">5</mn></msup><mo stretchy="false" is="true">)</mo></math> non-singular cells, including the complete adjacency information between these cells. This is done by a projection approach. We construct a special planar arrangement AS<math><msub is="true"><mi mathvariant="script" is="true">A</mi><mi is="true">S</mi></msub></math> with fewer cells than a cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. Based on ΩS<math><msub is="true"><mi is="true">Ω</mi><mi is="true">S</mi></msub></math> we also compute a simplicial complex which is isotopic to S. A complete C++-implementation of the stratification algorithm is presented. It shows good performance for many well-known examples from algebraic geometry.