  In this paper we consider monomial localizations of monomial ideals and
conjecture that a monomial ideal is polymatroidal if and only if all its
monomial localizations have a linear resolution. The conjecture is proved for
squarefree monomial ideals where it is equivalent to a well-known
characterization of matroids. We prove our conjecture in many other special
cases. We also introduce the concept of componentwise polymatroidal ideals and
extend several of the results, known for polymatroidal ideals, to this new
class of ideals.
