Combinatorics of 4-dimensional Resultant Polytopes

A. DICKENSTEIN; I. EMIRIS; V. FISIKOPOULOS

Boston

Congreso; ISSAC 2013; 2013

ACM

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 and up to dimension 3. 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 suciently generic, namely full dimensional and without parallel edges. Lastly, we oer a classication result of all possible 4-dimensional resultantpolytopes.