  An open bosonic string is considered with the aim to construct a general
gauge invariant, being a polynomial of Fubini-Veneziano (FV) fields. The FV
fields are transformed as 1-forms on $S^1$, that allows to formulate the
problem in geometric terms. We introduce a most general anzats for these
invariants and explicitly resolve the invariance conditions in the framework of
the anzats. The invariants are interpreted as integrals of n-form over a gauge
invariant domains in an n-dimensional torus, where the invariance of these
domains is considered with respect to the action of the diagonal of the group
$\times (Diff~S^1)^n$. We also discuss a possibility to get a complete set of
gauge invariants which allow an actual dependence on the string zero modes. We
find that the complete set can't be restricted by polynomial invariants only.
The classical polynomial invariants, being directly defined in the string Fock
space, turn out to break the structure of the respective BRST cohomology even
in the critical dimension. We discuss a possibility to restore the BRST
invariance of the corresponding operator algebra by a non-trivial quantum
deformation of the original invariants.
