In the first half of this paper, we complement the theory on
discrete polymatroids. More precisely,
(i) we prove that
a discrete polymatroid satisfying the strong exchange property is,
up to an affinity, of Veronese type;
(ii) we classify all uniform matroids which are level;
(iii) we introduce
the concept of ideals of fiber type and show that all
polymatroidal ideals are of fiber type.
On the other hand,
in the latter half of this paper,
we generalize the result proved by Stefan Blum
that the defining ideal of the Rees ring of
a base sortable matroid possesses a quadratic Gröbner basis.
For this purpose we introduce the concept of
``$l$-exchange property'' and show
that a Gröbner basis of
the defining ideal of the Rees ring of an ideal $I$
can be determined and that $I$ is of fiber type
if $I$ satisfies the $l$-exchange property.
Ideals satisfying the $l$-exchange property
include
strongly stable ideals,
polymatroid ideals of base sortable discrete polymatroids,
ideals of Segre-Veronese type and certain ideals related to classical
root systems.