Monomial Bases and Polynomial System Solving

This paper addresses the problem of eecient construction of monomial bases for the coordinate rings of zero-dimensional varieties. Existing approaches rely on Grr ob-ner bases methods { in contrast, we make use of recent developments in sparse elimination techniques which allow us to strongly exploit the structural sparseness of the problem at hand. This is done by establishing certain properties of a matrix formula for the sparse resultant of the given polynomial system. We use this matrix construction to give a simpler proof of the result of Pedersen and Sturmfels 22] for constructing mono-mial bases. The monomial bases so obtained enable the eecient generation of multiplication maps in coordinate rings and provide a method for computing the common roots of a generic system of polynomial equations with complexity singly exponential in the number of variables and polynomial in the number of roots. We describe the implementations based on our algorithms and provide empirical results on the well-known problem of cyclic n-roots; our implementation gives the rst known upper bounds in the case of n = 10 and n = 11. We also present some preliminary results on root nding for the Stewart platform and motion from point matches problems in robotics and vision respectively.