Doubling coverings of algebraic hypersurfaces

Abstract

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.