  We calculate explicitly the variation $\delta T_c$ of the Bose-Einstein
condensation temperature $T_c$ induced by weak repulsive two-body interactions
to leading order in the interaction strength. As shown earlier by general
arguments, $\delta T_c/T_c$ is linear in the dimensionless product $an^{1/3}$
to leading order, where $n$ is the density and $a$ the scattering length. This
result is non-perturbative, and a direct perturbative calculation of the
amplitude is impossible due to infrared divergences familiar from the study of
the superfluid helium lambda transition. Therefore we introduce here another
standard expansion scheme, generalizing the initial model which depends on one
complex field to one depending on $N$ real fields, and calculating the
temperature shift at leading order for large $N$. The result is explicit and
finite. The reliability of the result depends on the relevance of the large $N$
expansion to the situation N=2, which can in principle be checked by systematic
higher order calculations. The large $N$ result agrees remarkably well with
recent numerical simulations.
