2017/08/02 by Friedland, Omer, Yomdin, Yosef
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.00831
A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ:B1→ Y of the unit ball in ℂn, which is extendible to the (say) four times larger concentric ball of B1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney's ball coverings of a domain W⊂ \mathbb Rn, introduced in [17] (compare [9]). We study doubling chains in the complement Y=ℂn∖ H of a complex algebraic hypersurface H of degree d in ℂn, and provide information on their length and other properties. Our main result is that any two points v1,v2 in a distance δ from H can be joined via a doubling chain in the complement Y=ℂn∖ H of length at most c1log (\fracc2δ) with explicit constants c1,c2 depending only on n and d. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.