Triangulations of polytopes and computational algebra

Let ${\cal A}$ be a point configuration. The graph of triangulations of ${\cal A},$ G$\sb{\cal A}$, is the graph whose vertices are the triangulations of ${\cal A}$ and two triangulations are adjacent when there is a geometric bistellar operation taking one triangulation into the other. Chapter one discusses algorithms and computer software developed to compute G$\sb{\cal A}.$ A well-known open problem is to decide whether G$\sb{\cal A}$ is always a connected graph. The main result of chapter one indicates that a disconnected example may exist. We found an example of a three dimensional configuration whose graph of triangulations has low connectivity. Gel'fand, Kapranov and Zelevinsky asked whether products of two simplices could have non-regular triangulations. In chapter two we present an affirmative answer to this open question. Chapter two also contains a detailed study of triangulations for other (0,1)-polytopes such as cubes and hypersimplices. Our techniques include the use of Grobner bases theory. Chapter three discusses Grobner bases for a family of arrangements of linear subspaces. This family has a strong relation to coloring problems in graph theory. We apply our results to the enumeration of vertex colorings. In Chapter four we present an algorithmic version of a theorem of G. Polya. Given a real homogeneous polynomial F, strictly positive in the non-negative orthant, Polya's theorem says that for a sufficiently large exponent p the coefficients of F($x\sb1,$...,$x\sb{n})\cdot(x\sb1 + {\cdot\cdot\cdot} + x\sb{n})\sp{p}$ are strictly positive. As an application of Polya's theorem we present a new algorithm for the decomposition of a strictly positive polynomial as the quotient of two sums of squares of polynomials (special case of the algorithmic version of Hilbert's 17th problem). We also discuss how Hilbert's 17th problem originated in the study of geometric constructions with marked ruler.