On the Newton Polytope of the Resultant

The study of Newton polytopes of resultants and discriminants has its orgin in the work of Gelfand, Kapranov, and Zelevinsky on generalized hypergeometric functions (see e.g., [8]). Central to this theory is the notion of the A-discriminant AA, which is the discriminant of a Laurent polynomial with specified support set A (see [6, 7]). Two main results of Gelfand, Kapranov, and Zelevinsky are concerned with their secondary polytope £(A). First, the vertices of this polytope are in bijection with the coherent triangulations of A, and, secondly, the secondary polytope H(A) approximates the Newton polytope of the Adiscriminant A^. It was observed in [6, Proposition 1.3.1] that resultants are special instances of A-discriminants, and this observation was used in [9] to give an explicit combinatorial description of the Newton polytope of the classical Sylvester resultant. Subsequent papers extended the theory of Gelfand, Kapranov and Zelevinsky into several different directions. In [11] the A-resultant was introduced, and its interpretation as the Chow form of an associated toric variety leads to a refined geometric understanding of the relationship between triangulations of A and monomials in AA. In [3] the concept of secondary polytopes was extended to the more geometric construction of fiber polytopes. Product formulas of Poisson type, first given for the A-discriminant in [6, §2F], were proved in [14] for general Chow forms, for the A-resultant, and for the sparse mixed resultant, The present paper continues this line of research, but it is self-contained. Our main result is a combinatorial construction of the Newton polytope N(R) of the sparse mixed resultant R. To define these terms, we let A0, A1, ... An C Zn be subsets which jointly span the affine lattice Zn, and card(Ai) =: mi. Then R. is the unique (up to scaling) irreducible polynomial in m := m0 + m1 + • • • + mn variables ci,a, which vanishes whenever the Laurent polynomials