  We study the minimum weight basis problem on matroid when elements' weights
are uncertain. For each element we only know a set of possible values (an
uncertainty area) that contains its real weight. In some cases there exist
bases that are uniformly optimal, that is, they are minimum weight bases for
every possible weight function obeying the uncertainty areas. In other cases,
computing such a basis is not possible unless we perform some queries for the
exact value of some elements.
  Our main result is a polynomial time algorithm for the following problem.
Given a matroid with uncertainty areas and a query cost function on its
elements, find the set of elements of minimum total cost that we need to
simultaneously query such that, no matter their revelation, the resulting
instance admits a uniformly optimal base. We also provide combinatorial
characterizations of all uniformly optimal bases, when one exists; and of all
sets of queries that can be performed so that after revealing the corresponding
weights the resulting instance admits a uniformly optimal base.
