The Newton Polytope of the Sparse Resultant

The Newton polytope of the sparse resultant (or resultant polytope) offers a complexity measure of the resultant polynomial in sparse elimination. It is also useful for interpolation, in particular of the implicit equation of a parametric hypersurface. In this talk, we focus on 4-dimensional resultant polytopes, characterize all possible classes, and derive upper and lower bounds on their face numbers. This work extends results on the univariate case (Gelfand, Kapranov, Zelevinsky, 1990), and on 3-dimensional resultant polytopes (Sturmfels 1994).