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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