  A critical dark matter halo mass ($M_{\rm crit}$) for Population III (Pop
III) stars can be defined as the typical minimum halo mass that hosts
sufficient cold dense gas required for the formation of the first stars. The
presence of Lyman-Werner (UV) radiation, which can dissociate molecular
hydrogen, and the baryon-dark matter streaming velocity both delay the
formation of Pop III stars by increasing $M_{\rm crit}$. In this work, we
constrain $M_{\rm crit}$ as a function of Lyman-Werner flux (including
self-shielding), baryon-dark matter streaming, and redshift using cosmological
simulations with a large sample of halos utilizing the adaptive mesh refinement
(AMR) code ENZO. We provide a fit for $M_{\rm crit}$ as a function of these
quantities which we expect to be particularly useful for semi-analytical models
of early galaxy formation. In addition, we find: (i) the measured redshift
dependence of $M_{\rm crit}$ in the absence of radiation or streaming is
$(1+z)^{-1.58}$, consistent with a constant virial temperature; (ii) increasing
the UV background increases $M_{\rm crit}$ while steepening the redshift
dependence, up to $(1+z)^{-5.7}$; (iii) baryon-dark matter streaming boosts
$M_{\rm crit}$ but flattens the dependence on redshift; (iv) the combination of
the two effects is not simply multiplicative.
