Computational Synthetic Geometry

Jürgen Bokowski, Bernd Sturmfels (auth.)3540504788, 9783540504788, 0387504788

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.

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Table of contents :
Preliminaries….Pages 1-17
On the existence of algorithms….Pages 18-31
Combinatorial and algebraic methods….Pages 32-60
Algebraic criteria for geometric realizability….Pages 61-86
Geometric methods….Pages 87-101
Recent topological results….Pages 102-114
Preprocessing methods….Pages 115-132
On the finding of polyheadral manifolds….Pages 133-146
Matroids and chirotopes as algebraic varieties….Pages 147-157