  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 $d_1, \ldots, d_m$ 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 $d_1,
\ldots , d_m$, 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.
