Computing the Newton Polytope of Specialized Resultants

We consider sparse (or toric) elimination theory in order to describe, by combinatorial means, the monomials appearing in the (sparse) resultant of a given overconstrained algebraic system. A modification of reverse search allows us to enumerate all mixed cell configurations of the given Newton polytopes so as to compute the extreme monomials of the Newton polytope of the resultant. We consider specializations of the resultant to a polynomial in a constant number of variables (typically up to 3) and propose a combinatorial algorithm for computing its Newton polytope; our algorithm need only examine the silhoutte of the secondary polytope with respect to an orthogonal projection in a space of as many dimensions. We describe the Newton polygon of the implicit equation of a rational parametric curve in a self-contained manner by purely combinatorial arguments; the complexity of our method is almost linear in the cardinality of the supports of the parametric polynomials. We extend certain of these results to describing the Newton polytope of the implicit equation of a polynomial parametric surface. Classification: Algebraic geometry, Discrete geometry.