Multihomogeneous resultant matrices

Multihomogeneous structure in algebraic systems is the first step away from the classical theory of homogeneous equations towards fully exploiting arbitrary supports. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure B\'ezout types, including hybrid matrices. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman.

One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. 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 B\'ezout-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 an example showing all kinds of matrices that may be encountered, and illustrations of our Maple implementation.

This is joint work with A. Dickenstein.