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

This paper studies generic and perturbation properties inside the linear space of polynomial matrices whose rows have degrees bounded by a given list of natural numbers, which in the particular case is just the set of 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 [29] only for polynomial matrices with degree at most d. Surprisingly, most of the properties proved in [29], 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 , and with right minimal indices differing at most by one and having a sum equal to , and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.