Combinatorics of 4-dimensional resultant polytopes

The Newton polytope of the resultant, or resultant polytope, characterizes the resultant polynomial more precisely than total degree. The combinatorics of resultant polytopes are known in the Sylvester case [Gelfand et al.90] and up to dimension 3 [Sturmfels 94]. We extend this work by studying the combinatorial characterization of 4-dimensional resultant polytopes, which show a greater diversity and involve computational and combinatorial challenges. In particular, our experiments, based on software respol for computing resultant polytopes, establish lower bounds on the maximal number of faces. By studying mixed subdivisions, we obtain tight upper bounds on the maximal number of facets and ridges, thus arriving at the following maximal f-vector: (22,66,66,22), i.e. vector of face cardinalities. Certain general features emerge, such as the symmetry of the maximal f-vector, which are intriguing but still under investigation. We establish a result of independent interest, namely that the f-vector is maximized when the input supports are sufficiently generic, namely full dimensional and without parallel edges. Lastly, we offer a classification result of all possible 4-dimensional resultant polytopes.