  We consider the effects of a bare mass term for the inflaton, when the
inflationary potential takes the form $V(\phi)= \lambda \phi^k$ about its
minimum with $k \ge 4$. We concentrate on $k=4$, but discuss general cases as
well. Further, we assume $\lambda \phi_{\rm end}^2 \gg m_\phi^2$, where
$\phi_{\rm end}$ is the inflaton field value when the inflationary expansion
ends. We show that the presence of a mass term (which may be present due to
radiative corrections or supersymmetry breaking) can significantly alter the
reheating process, as the equation of state of the inflaton condensate changes
from $w_\phi=\frac{1}{3}$ to $w_\phi=0$ when $\lambda \phi^2$ drops below
$m_\phi^2$. We show that for a mass $m_\phi \gtrsim T_{\rm RH}/250$, the mass
term will dominate at reheating. We compute the effects on the reheating
temperature for cases where reheating is due to inflaton decay (to fermions,
scalars, or vectors) or to inflaton scattering (to scalars or vectors). For
scattering to scalars and in the absence of a decay, we derive a strong upper
limit to the inflaton bare mass $m_\phi < 350~{\rm MeV} (T_{\rm
RH}/10^{10}~{\rm GeV})^{3/5}$, as there is always a residual inflaton
background which acts as cold dark matter. We also consider the effect of the
bare mass term on the fragmentation of the inflaton condensate.
