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 (J. Algebra, 163 (1994) 115; J. Algebraic Geom., 3 (1994) 569). 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.