Multihomogeneous resultant formulae by means of complexes

Abstract

The first step in the generalization of the classical theory of homogeneous equations to the case of arbitrary support is to consider algebraic systems with multihomogeneous structure. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, and including matrices of hybrid type of these two. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman [SZ94], [WZ94]. One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. Whenever such formulae exist, we specify the underlying complexes so as to make the resultant matrix explicit. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bézout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with examples showing the kinds of matrices that may be encountered, and illustrations of our Maple implementation.

Introduction to residues and resultants

“…For a modern account of determinantal formulas for multihomogeneous resultants see [DE03]. Multihomogeneous resultants are special instances of sparse (or toric) resultants.…”

Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

“…The are many papers and books on the theory and computation of resultants for a variety of different types of polynomial systems, e.g., [13] (see Chapters 3 and 7), [18] (see Chapters 3, 8, and 13), and [16,17,25].…”

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

“…The possible values of (m1, m2) which lead to determinantal complexes is a finite set [21,26]. Many classically known resultant formulas can be obtained as the determinant of different instances of φ in (3), for particular integers (m1, m2) ∈ Z 2 .…”

“…[49]). General such formulas for unmixed multilinear systems are identified in [21,26,45,49]. In general, the degree of φ : Kν,q → K ν−1,q with respect to f0, f1, f2 is equal to (q − q + 1)+ therefore degree one linear maps arise from q = q .…”

“…Many, if not all, classical resultant matrices are instances of such complexes [49], including the projective resultant [19]. In [21,26] there is a systematic exploration of possible determinantal complexes, and also a software package that produces formulas for unmixed (and even scaled) resultants. Interestingly, there is a plethora of hybrid resultant matrices that consist of Bézout-type and Sylvester-type blocks [21,26], see also [17].…”