Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry

Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation Theory, and Computational Geometry. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the paper. We present also some new results, stressing interconnection between various types and various applications of smooth parametrization

Bautin ideals and Taylor domination

This research was supported by the ISF, grant no. 639/09 and by the Minerva foundation

Cauchy Type Integrals of Algebraic Functions

We consider Cauchy type integrals $I(t)={1\over 2\pi i}\int_{\gamma} {g(z)dz\over z-t}$ with $g(z)$ an algebraic function. The main goal is to give constructive (at least, in principle) conditions for $I(t)$ to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the Monodromy group of the algebraic function $g$, the geometry of the integration curve $\gamma$, and the analytic properties of the Cauchy type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincar\'e Center-Focus problem and the second part of the Hilbert 16-th problem.Comment: 58 pages, 19 figure

Bautin ideals and Taylor domination

This research was supported by the ISF, grant no. 639/09 and by the Minerva foundation