[Paper Review] Doubling coverings of algebraic hypersurfaces

Omer Friedland, Yosef Yomdin|arXiv (Cornell University)|Dec 9, 2015

Analytic Number Theory Research20 references3 citations

TL;DR

This paper establishes an upper bound of order log(1/δ) on the minimal number of charts in doubling coverings of compact parts of non-singular algebraic hypersurfaces Y = {P = c}, where δ is the distance from Y to the singular set of a polynomial P with non-degenerate critical points. The result shows that analytic doubling coverings become increasingly complex as the hypersurface approaches singularities, in contrast to C^k-parameterizations, which remain uniformly bounded in complexity.

ABSTRACT

A doubling covering $\U$ of a complex $n$-dimensional manifold $Y$ consists of analytic functions $ψ_j:B_1 o Y$, 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 $κ({\U})$ 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 $κ({\U})$ is of order $\log({1}/ρ)$, where $ρ$ 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 $Ω\subset G$ in $Y$. We shortly indicate connections with Kobayashi metric and with Harnack inequality.

Motivation & Objective

• To establish an upper bound on the minimal number of charts in doubling coverings of compact parts of non-singular algebraic hypersurfaces in C^n.
• To investigate how the complexity of such coverings grows as the hypersurface approaches the singular set of the defining polynomial P.
• To connect doubling coverings with doubling inequalities for analytic functions on hypersurfaces.
• To demonstrate that analytic doubling coverings exhibit logarithmic complexity growth near singularities, contrasting with uniform boundedness in C^k-parameterizations.

Proposed method

• Use of doubling coverings defined by univalent analytic maps ψ_j: B_1 → Y, extendable to B_4, to cover compact domains in Y.
• Application of the main result (Theorem 3.1) to bound the number of charts in terms of the distance δ to the singular set of P.
• Derivation of doubling inequalities for analytic functions f on Y via chains of charts in doubling coverings, analogous to Harnack-type constructions.
• Use of the norm of the gradient ∇P to estimate the distance from Y to the singular locus, with K(P) = ||∇P|| defining a scale factor.
• Construction of explicit coverings for special cases like hyperbolas H_ε = {zy = ε²} and quadrics Y_ε = {∑z_j² = ε²}, showing matching upper and lower bounds.
• Application of Corollary 5.3 to derive lower bounds on chart count using the doubling constant DC_f(G,Ω) of analytic functions.

Experimental results

Research questions

• RQ1 How does the minimal number of charts in a doubling covering of a compact part of a non-singular algebraic hypersurface Y = {P = c} grow as the hypersurface approaches the singular set of P?
• RQ2 Can doubling coverings provide a complexity measure that captures the geometric blow-up near singularities, unlike C^k-parameterizations?
• RQ3 What is the precise dependence of the covering complexity on the distance δ to the singular set for polynomials with non-degenerate critical points?
• RQ4 How are doubling coverings related to doubling inequalities for analytic functions on hypersurfaces?
• RQ5 Can the logarithmic complexity bound be extended to polynomials with degenerate or non-isolated singularities?

Key findings

• The minimal number of charts in a doubling covering of G_c = Y_c ∩ Q is bounded above by C(P) log(1/δ), where δ is the distance from Y_c to the critical points of P.
• For the hyperbola H_ε = {zy = ε²}, the number of charts in any doubling covering with ρ ≥ 1/10 is at least c₃ log(1/ε), matching the upper bound.
• In the case of the quadric Y_ε = {∑z_j² = ε²}, the number of charts is bounded by c₅ log(c₆/δ), with constants depending only on n.
• Doubling inequalities for analytic functions f on Y are shown to be controlled by chains of charts in doubling coverings, with explicit dependence on the doubling constant DC_f(G,Ω).
• The construction yields a sharp logarithmic complexity growth for doubling coverings near singularities, in contrast to the uniform boundedness of C^k-parameterizations.
• The results suggest that doubling coverings are a natural framework for studying complexity in smooth dynamics and diophantine geometry, particularly in relation to entropy and rational point density.