Computational synthetic geometry / Jürgen Bokowski, Bernd Sturmfels

This work concerns the realization of combinatorial structures, primarily over the reals or other ordered fields, but also over the rationals, complex numbers or other fields. Both theoretical and algorithmic aspects are treated. The combinatorial structures considered include abstract polytopes, incidence structures, matroids and oriented matroids. Of course, the realization problems for these combinatorial objects are interrelated. The approach in this book involves invariant theory and algebraic geometry, as the above problems are reduced to finding solutions to the Grassmann-Plücker relations which take into account the combinatorial structure. One of the main contributions described is the final polynomial, a polynomial in the brackets (or determinants) which immediately displays the nonrealizability of a structure.
This book is a readable and interesting introduction to the authors' important contributions to the algebraic and invariant-theoretic side of computational geometry. (MathSciNet)