  Sparse (or toric) elimination exploits the structure of polynomials by
measuring their complexity in terms of Newton polytopes instead of total
degree. The sparse, or Newton, resultant generalizes the classical homogeneous
resultant and its degree is a function of the mixed volumes of the Newton
polytopes. We sketch the sparse resultant constructions of Canny and Emiris and
show how they reduce the problem of root-finding to an eigenproblem. A novel
method for achieving this reduction is presented which does not increase the
dimension of the problem. Together with an implementation of the sparse
resultant construction, it provides a general solver for polynomial systems. We
discuss the overall implementation and illustrate its use by applying it to
concrete problems from vision, robotics and structural biology. The high
efficiency and accuracy of the solutions suggest that sparse elimination may be
the method of choice for systems of moderate size.

Symbolic Computation;Numerical Analysis;Commutative Algebra