Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry

TL;DR

This paper reviews the concept of smooth parametrization across various mathematical fields, highlighting similarities, open problems, and new results, to unify understanding and explore interconnections among these domains.

Contribution

It provides an overview of smooth parametrization results, open problems, and introduces new findings linking different applications across multiple mathematical disciplines.

Findings

Identifies structural similarities across domains

Highlights open problems and conjectures

Introduces new results connecting various applications

Abstract

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.