Robustness and perturbations of minimal bases II: The case with given row degrees

Abstract

This paper studies generic and perturbation properties inside the linear space of m ×(m +n) polynomial matrices whose rows have degrees bounded by a given list d1, ..., dm of natural numbers, which in the particular case d1=···=dm =d is just the set of m ×(m +n) polynomial matrices with degree at most d. Thus, the results in this paper extend to a much more general setting the results recently obtained in "Robustness and perturbations of minimal bases" (2018) only for polynomial matrices with degree at most d. Surprisingly, most of the properties proved in "Robustness and perturbations of minimal bases" (2018), as well as their proofs, remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well-known Sylvester matrices of polynomial matrices are replaced by the new trimmed Sylvester matrices introduced in this paper. The following results are presented, among many others, in this work: (1) generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d1, ..., dm, and with right minimal indices differing at most by one and having a sum equal to ∑mi=1di, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.