Deformation Techniques for Sparse Systems

Abstract

We exhibit a probabilistic symbolic algorithm for solvingzero-dimensional sparse systems. Our algorithm combines a symbolichomotopy procedure, based on a flat deformation of a certainmorphism of affine varieties, with the polyhedral deformation ofHuber and Sturmfels. The complexity of our algorithm is cubic inthe size of the combinatorial structure of the input system. Thissize is mainly represented by the cardinality and mixed volume ofNewton polytopes of the input polynomials and an arithmeticanalogue of the mixed volume associated to the deformations underconsideration.