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.