On Functions and Curves Defined by Ordinary Differential Equations

These notes constitute a substantially extended version of a talk given in the Fields Institute (Toronto) during the semester \Singularities and Geometry", that culminated by Arnoldfest in celebration of V. I. Arnold's 60th anniversary. We give a survey of diierentresults showing how an upper bound for the numberof isolated zeros for functions satisfying ordinary diierential equations, may be obtained without solving these equations. The main source of applications is the problem on zeros of complete Abelian integrals, one of the favorite subjects discussed on Arnold's seminar in Moscow for over quarter a century. Data quatione quotcunque uentes quantitt invol-vente uxiones invenire et vice versa. It is useful to solve diierential equations. Translation by Vladimir Arnold x1. Introduction 1.1. Equations and solutions. One of the illusions that are pleasant to nourish is the claim that simple equations cannot have complicated solutions. Though completely refuted by the recent progress in the dynamical systems, this principle still holds in a more restricted context. For example, a planar real algebraic curve of some known degree d cannot have too many real ovals on the real plane and cannot intersect straight lines by too many (more than d) isolated points. This example can be easily generalized to algebraic varieties of higher dimensions. Thus at least in the context of elementary real or complex algebraic geometry simple descriptions cannot lead to perverse objects. The requirement of algebraicity is too restrictive, as was relatively recently discovered by A. Khovanskii Kh]. One can in fact allow all elementary functions