|Abstract||Several combinatorial aspects of triangulations and their generalizations are studied in this thesis. A triangulation of a point configuration and a d-dimensional polyhedron whose vertices are among the points is a decomposition of the polyhedron using d-simplices with vertices|
among the points.
The two main fields triangulations appear are combinatorial geometry in mathematics and computational geometry in information science. The topics connected to triangulations in combinatorial geometry include, polytope theory, Grobner bases of affine toric ideals, Hilbert bases, generalized hypergeometric functions, and Ehrhart polynomials. Many fields of computational geometry, such as computer graphics, solid modeling, mesh generation, and motion planning, use triangulations extensively. The problems we consider are among the main interests in combinatorial geometry.