  We investigate the dependence of the peak of the IMF on the physics of the
so-called first Larson core, which corresponds to the point where the dust
becomes opaque to its own radiation. We perform numerical simulations of
collapsing clouds of $1000 M_\odot$ for various gas equation of state (eos),
paying great attention to the numerical resolution and convergence. The initial
conditions of these numerical experiments are varied in the companion paper. We
also develop analytical models that we confront to our numerical results. If an
isothermal eos is used, we show that the peak of the IMF shifts to lower masses
with improved numerical resolution. When an adiabatic eos is employed,
numerical convergence is obtained. The peak position varies with the eos and we
find that the peak position is about ten times the mass of the first Larson
core. By analyzing the stability of non-linear density fluctuations in the
vicinity of a point mass and then summing over a reasonable density
distribution, we find that tidal forces exert a strong stabilizing effect and
likely lead to a preferential mass several times larger than that of the first
Larson core. We propose that in a sufficiently massive and cold cloud, the peak
of the IMF is determined by the thermodynamics of the high density adiabatic
gas as well as the stabilizing influence of tidal forces. The resulting
characteristic mass is about ten times the mass of the first Larson core, which
altogether leads to a few tenths of solar masses. Since these processes are not
related to the large scale physical conditions and to the environment, our
results suggest a possible explanation for the apparent universality of the
peak of the IMF.
