  We derive exact renormalization-group equations for the $n$-point vertices
($n=0,1,2,\cdots$) of interacting single-component Bose-Einstein condensates
based on the vertex expansion of the effective action. They have a notable
feature of automatically satisfying Goldstone's theorem (I), which yields the
Hugenholtz-Pines relation $\Sigma(0)-\mu=\Delta(0)$ as the lowest-order
identity. Using them, it is found that the anomalous self-energy $\Delta(0)$
vanishes below $d_{\rm c}=4$ ($d_{\rm c}=3$) dimensions at finite temperatures
(zero temperature), contrary to the Bogoliubov theory predicting a finite
"sound-wave" velocity $v_{\rm s}\propto [\Delta(0)]^{1/2}>0$. It is also argued
that the one-particle density matrix $\rho({\bf
r})\equiv\langle\hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$
for $d<d_{\rm c}$ dimensions approaches the off-diagonal-long-range-order value
$N_{\bf 0}/V$ asymptotically as $r^{-d+2-\eta}$ with an exponent $\eta>0$. The
anomalous dimension $\eta$ at finite temperatures is predicted to behave for
$d=4-\epsilon$ dimensions ($0<\epsilon\ll 1$) as $\eta\propto\epsilon^2$. Thus,
the interacting Bose-Einstein condensates are subject to long-range
fluctuations similar to those at the second-order transition point, and their
excitations in the one-particle channel are distinct from the Nambu-Goldstone
mode with a sound-wave dispersion in the two-particle channel.
