  We extend the SU(3) (Polyakov) Nambu Jona-Lasinio in two ways: We introduce
the next to leading order contribution (in $N_c$) in the partition function.
This contribution contains explicit mesonic terms. We introduce a coupling
between the gluon field and the quark degrees of freedom which goes beyond a
simple rescaling of the critical temperature. With both these improvements we
can reproduce, for vanishing chemical potentials, the lattice results for the
thermal properties of a strongly interacting system like pressure, energy
density, entropy density, interaction measure and the speed of sound. Also the
expansion parameter towards small but finite chemical potentials agrees with
the lattice results. Extending the calculations to finite chemical potentials
(what does not require any new parameter) we find a first order phase
transition up to a critical end point of $T_{CEP}= 110\ MeV$ and $\mu_q = 320\
MeV$. For very large chemical potentials, we find agreement with pQCD
calculations. We calculate the mass of mesons and baryons as a function of
temperature and chemical potential and the transition between the hadronic and
the chirally restored phase. These calculations provide an equation of state in
the whole $T,\mu$ plane an essential ingredient for dynamical calculations of
ultra-relativistic heavy ion collisions but also for the physics of neutron
stars and neutron star collisions.
