The resultant of two polynomials is an expression constructed from those polynomials such that it is zero precisely when the two polynomials share a root. Because of its desirable properties, the resultant is frequently used in fields such as number theory and computer algebra. Itself a polynomial, the resultant can be studied using all of the same tools of analysis used for any polynomial. One of these tools is the Newton polytope, which is the convex hull of the exponent vector of a polynomial (equivalently, a set of monomials), and which gives insight into both the zeros of the polynomial and its irreducibility. In this paper, the authors examine the combinatorics of the 4-dimensional resultant polytope, which is the Newton polytope of a resultant. Among their results, they derive bounds on the number of features these resultant polytopes can have.
The paper establishes its bounds through a mix of experiment and theory. Empirical results, using the software library ResPol, establish lower bounds on the number of faces, while an approach based on mixed subdivisions yields upper bounds for facets and ridges (22 and 66, respectively). In a rather surprising result, the authors also provide a loose upper bound on the number of vertices at 28 from the previously known bound of 6608. The work builds to their main theorem, which provides a general characterization of the combinatorics of 4-dimensional resultant polytopes.
The paper is quite well written and very thorough, with tightly crafted and succinct language. It is also a dense work, and a reader not already familiar with the particular topic will need to spend time digesting the information. This is not a failing on the part of the authors; the paper is a conference publication, and the target audience is precisely those who are (presumably) already familiar with the topic.