GRAPE

GRAPE is about the use of polyhedral methods in the theory of sparse polynomial systems. The goal is to combine polyhedral and toric geometry with the study of sparse systems and especially concerning resultants and Gröbner bases. In the Turkish side of the project we focus on the polyhedral aspects. In particular we generalize the notion of Newton polytope to that of Newton fan. Instead of considering all polynomials having a given Newton polytope, we consider all polynomials whose Newton polytopes have the same normal fan. Then, we study a multivariate version of Hilbert series and connect to the generating functions of linear Diophantine systems. Finally, in the context of toric geometry, we study the Hilbert series and other characteristics of ideals coming from graphs. Of particular interest is the study of Newton polytopes of edge rings. The main goal of the international collaboration is the derivation of better algorithms and complexity bounds for sparse resultants and Gröbner bases. Last year, members of the French team presented a first step towards this goal. It is clear that more polyhedral geometry is needed in order to proceed and this is what this project is about. Bringing together two complementary teams in order to work on a state-of-the-art topic. But instead of trying to improve the existing result only, what we suggest is a generalization of the framework as well.