Subdivisions for macaulay formulas of sparse systems

Abstract

In a seminal article [7], D'Andrea describes a method for determining Macaulay-type formulae for the resultants of sparse polynomial systems. His algorithm works recursive, reducing the dimension n of the problem at each step. In doing do, he applies a certain coherent mixed subdivision of the given Newton polytopes into cells, each representing a system with smaller dimension. To simplify this procedure, we insert an intermediate step in which these reduced systems are transferred to the n-dimensional domain of the complete cells. As a consequence, the input system of each iteration step need not contain an additional polytope and only one system per secondary cell has to be considered. The individual subdivisions determined in various steps of the algorithm are combined into a single subdivision of the whole problem. Only then, the matrix for calculating the resultant is determined. To prove our method, we generalize a theorem of [22] on the initial form of resultants with respect to coherent mixed subdivisions.