Regular triangulations and resultant polytopes

We describe properties of the Resultant polytope of a given set of polynomial equations towards an outputsensitive algorithm for enumerating its vertices. In principle, one has to consider all regular fine mixed subdivisions of the Minkowski sum of the Newton polytopes of the given equations. By the Cayley trick, this is equivalent to computing all regular triangulations of another point set in higher dimension. However, the number of all regular triangulations is generally much larger than that of the vertices of the Resultant polytope, as illustrated by our experiments [3]. Thus, we study output-sensitive methods by defining classes of subdivisions, called configurations, which yield the same resultant vertex. Moreover, we offer algorithmic versions of certain results by Sturmfels [11], regarding the edges of the Resultant polytope. Lastly, we settle some easy cases, and discuss harder examples.