  The (left) linear hull of a weighted automaton over a field is a topological
invariant. If the automaton is minimal, the linear hull can be used to
determine whether or not the automaton is equivalent to a deterministic one.
Furthermore, the linear hull can also be used to determine whether the minimal
automaton is equivalent to an unambiguous one. We show how to compute the
linear hull, and thus prove that it is decidable whether or not a given
automaton over a number field is equivalent to a deterministic one. In this
case we are also able to compute an equivalent deterministic automaton. We also
show the analogous decidability and computability result for the unambiguous
case. Our results resolve a problem posed in a 2006 survey by Lombardy and
Sakarovitch.
