  We present an accurate investigation of the algebraic conditions that the
symbols of a univariate, binary, Hermite subdivision scheme have to fulfil in
order to reproduce polynomials. These conditions are sufficient for the scheme
to satisfy the so called spectral condition. The latter requires the existence
of particular polynomial eigenvalues of the stationary counterpart of the
Hermite scheme. In accordance with the known Hermite schemes, we here consider
the case of a Hermite scheme dealing with function values, first and second
derivatives. Several examples of application of the proposed algebraic
conditions are given in both the primal and the dual situation.
