Quantitative Theory of Ordinary Differential Equations and Tangential Hilbert 16th Problem

These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples illustrating various phenomena related to this problem. Sometimes these examples give an insight concerning the proofs, though the complete exposition of the latter is mostly relegated to separate expositions. Lecture 1. Hilbert 16th problem: Limit cycles, cyclicity, Abelian integrals In the first lecture we discuss several possible relaxed formulations of the Hilbert 16th problem on limit cycles of vector fields and related questions from analytic functions theory. 1. Zeros of analytic functions The introductory section presents several possible formulations of the question about the number of zeros of a function of one variable. All functions below are either real or complex analytic in their domains, eventually exhibiting singularities on their boundaries. We would like to stress that only isolated zeros of such functions are counted, so that by definition a function identically vanishing on an open set, has no isolated zeros there. Exposition goes mostly by examples that are separated from each other by the symbol ◭. A few demonstrations terminate by the usual symbol . 1.1. Nonaccumulation and individual finiteness. A function f(t) real analytic on a finite open interval (a, b) ⊂ R may have an infinite number of isolated zeros on this interval only if they accumulate to the boundary points a, b of the latter. Thus the finiteness problem of decision whether or not the given function f has only finitely many zeros in its domain, is reduced to studying the boundary behavior of f . In particular, if f is analytic also at the boundary points a, b, then accumulation of infinitely many zeros to these points is impossible and hence f has only finitely many roots on the interval (a, b). However, this strong condition of analyticity can be relaxed very considerably. 1991 Mathematics Subject Classification. [. The research was supported by the Israeli Science Foundation grant no. 18-00/1. c ©0000 (copyright holder)