Multihomogeneous resultant formulae for systems with scaled support

摘要

Constructive methods for matrices of multihomogeneous resultants for unmixed systems have been studied in [7, 13, 15]. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for the corresponding degree vectors, as well as precise domains where these concentrate; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, in the general case. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, which we call partial Bezoutians. Our public-domain Mapleimplementation includes efficient storage of complexes in memory, and construction of resultant matrices.