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 $m\times (m+n)$ polynomial matrices whose rows have degrees bounded by a given list $d_1,\dots ,d_m$ of natural numbers, which in the particular case $d_1=\cdots =d_m=d$ is just the set of $m\times (m+n)$ 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 [Van Dooren & Dopico, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.011] only for polynomial matrices with degree at most $d$. Surprisingly, most of the properties proved in [Van Dooren & Dopico, Linear Algebra Appl. (2017)], 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 $d_1,\dots ,d_m$, and with right minimal indices differing at most by one and having a sum equal to ${\sum }_{i=1}^m{d}_i$, and (2), under perturbations, these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly.