Computational Synthetic Geometry

1. Front Matter

   Pages I-IV

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2. Preliminaries

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 1-17

3. On the existence of algorithms

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 18-31

4. Combinatorial and algebraic methods

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 32-60

5. Algebraic criteria for geometric realizability

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 61-86

6. Geometric methods

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 87-101

7. Recent topological results

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 102-114

8. Preprocessing methods

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 115-132

9. On the finding of polyhedral manifolds

   • Jürgen Bokowski, Bernd Sturmfels

   Pages 133-146

10. Matroids and chirotopes as algebraic varieties

    • Jürgen Bokowski, Bernd Sturmfels

    Pages 147-157

11. Back Matter

    Pages 158-168

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Computational synthetic geometry deals with methods for realizing abstract geometric objects in concrete vector spaces. This research monograph considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It turns out that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems a variety of symbolic algorithms are discussed, and the methods are applied to obtain new mathematical results on convex polytopes, projective configurations and the combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and oriented matroids are introduced providing a new basis for applying computer algebra methods in this field. The necessary background knowledge is reviewed briefly. The text is accessible to students with graduate level background in mathematics, and will serve professional geometers and computer scientists as an introduction and motivation for further research.

• Mathematica
• Microsoft Access
• Processing
• Vector space
• boundary element method
• complexity
• computer algebra
• construction
• discrete geometry
• eXist
• field
• knowledge
• manifold
• mathematics
• theorem

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