  Motivated by the recent detection of metals in different components of the
high redshift universe and by the abundance ratios measured in the extremely
metal-poor stars of our Galaxy, we study the nucleosynthesis constraints that
this imposes on an early generation of stars (Population III). To do so we take
into account the chemical yields obtained from homogeneous evolutionary
calculations of zero metal stars in the mass range $3\la m/M_\odot\la 40$
(Limongi et al. 2000, Chieffi et al. 2001). We also consider the role played by
metal-free very massive objects (m$>100$ M$_\odot$). Using both analytical and
numerical chemical evolution models, we confront model predictions from the
different choices of the mass function proposed for Population III with the
observational constraints. We show that low values of star formation efficiency
($<1%$) are required so as not to exceed the minimum metallicity ([C/H]$\approx
-2.4$) measured in the high redshift systems for any of the IMFs proposed. We
also show that the observational constraints require $\Omega_{sr}< 3\times
10^{-3}\Omega_b$, confirming previous claims that the possible contribution of
the stellar remnants from Population III to the baryonic dark matter is
insignificant. At present, however, the scarcity of abundance measurements for
high redshift systems does not permit us to put severe limitations on the
nature of the initial mass function for Population III. In fact, overabundances
of alpha-elements with respect to iron of the order of those measured in damped
Lyman-$\alpha$ systems are obtained for any of the IMFs tested. Nevertheless,
to account for the very large [C,N/Fe] ratios found in a considerable number of
extremely metal-poor stars of our Galaxy, an IMF peaking at the intermediate
stellar mass range (4-8 M$_\odot$) is needed.
