  We introduce weighted one-deterministic-counter automata (ODCA). These are
weighted one-counter automata (OCA) with the property of counter-determinacy,
meaning that all paths labelled by a given word starting from the initial
configuration have the same counter-effect. Weighted ODCAs are a strict
extension of weighted visibly OCAs, which are weighted OCAs where the input
alphabet determines the actions on the counter.
  We present a novel problem called the co-VS (complement to a vector space)
reachability problem for weighted ODCAs over fields, which seeks to determine
if there exists a run from a given configuration of a weighted ODCA to another
configuration whose weight vector lies outside a given vector space. We
establish two significant properties of witnesses for co-VS reachability: they
satisfy a pseudo-pumping lemma, and the lexicographically minimal witness has a
special form. It follows that the co-VS reachability problem is in P.
  These reachability problems help us to show that the equivalence problem of
weighted ODCAs over fields is in P by adapting the equivalence proof of
deterministic real-time OCAs by B\"ohm et al. This is a step towards resolving
the open question of the equivalence problem of weighted OCAs. Furthermore, we
demonstrate that the regularity problem, the problem of checking whether an
input weighted ODCA over a field is equivalent to some weighted automaton, is
in P. Finally, we show that the covering and coverable equivalence problems for
uninitialised weighted ODCAs are decidable in polynomial time. We also consider
boolean ODCAs and show that the equivalence problem for (non-deterministic)
boolean ODCAs is in PSPACE, whereas it is undecidable for (non-deterministic)
boolean OCAs.
