  A doubling covering $\U$ of a complex $n$-dimensional manifold $Y$ consists
of analytic functions $\psi_j:B_1\to Y$, each function being analytically
extendable, as a mapping to $Y$, to a four times larger concentric ball $B_4$.
  Main result of this paper is an upper bound on the minimal number
$\kappa({\U})$ of charts in doubling coverings of a manifold $Y$, being a
compact part of a non-singular level hypersurface $Y=\{P=c\}$, where $P$ is a
polynomial on $\C^n$ with non-degenerated critical points. We show that
$\kappa({\U})$ is of order $\log({1}/{\rho})$, where $\rho$ is the distance
from $Y$ to the singular set of $P$.
  Our main motivation is that doubling coverings form a special class of
"smooth parameterizations", which are used in bounding entropy type invariants
in smooth dynamics on one side, and in bounding density of rational points in
diophantine geometry on the other. Complexity of smooth parameterizations is a
key issue in some important open problems in both areas.
  We also present connections between doubling coverings and doubling
inequalities for analytic functions $f$ on $Y$, which compare the maxima of
$|f|$ on couples of compact domains $\Omega\subset G$ in $Y$. We shortly
indicate connections with Kobayashi metric and with Harnack inequality.
