Hilbert 16th Problem and Related Topics in Complex Analysis and Foliations

Cornell University, Ithaca NY

Investigators

Abstract

Hilbert's sixteenth problem remains among the most persistent in his famous list, yielding first place in this regard only to the Riemann Hypothesis. This project focuses on Hilbert's sixteenth problem for quadratic vector fields, the infinitesimal version of Hilbert's sixteenth problem, and several related topics. The latter include the following: algebraic solutions of polynomial differential equations in higher dimensions; rigidity of complex polynomial foliations; relations between moduli of elliptic curves and rotation numbers; new problems about analytic families of germs of conformal maps, related to so-called mixed families. The first part of the project is based on two major results obtained under the principal investigator's previous award. The first of those results is an almost complete solution to the restricted Hilbert problem for quadratic vector fields. This solution is to be completed in the current project. In its final form, this solution would provide a "covering function" on the space of quadratic vector fields, a covering function with the following properties: (1) it is finite and lower semicontinuous on the set of vector fields that have no polycycles (separatrix polygons); (2) its restriction to this set has an infinite limit on the exclusive set of vector fields with polycycles; (3) it majorizes the number of limit cycles outside the exclusive set. The principal investigator proposes to find an upper bound on the number of limit cycles by establishing a local persistence property for limit cycles and then replacing the covering function with a cut-off function that still majorizes the number of limit cycles. The maximum of the cut-off function will be the desired upper bound. The second major objective of the current project is the complete solution of the restricted infinitesimal Hilbert sixteenth problem. This solution involves another covering function, this time on the set of all ultra-Morse polynomials of given degree, with the following property: it majorizes the number of real zeros of an Abelian integral of any polynomial one-form of degree smaller than the given one over the real ovals of a real ultra-Morse polynomial. The covering function has poles on the set of non-ultra-Morse polynomials. Once more, the goal is to prove a local persistence theorem for zeros of Abelian integrals and to derive an upper bound for the number of these zeros by replacing the covering function with a cut-off function. This will solve the infinitesimal Hilbert problem. The problems described in the latter part of the proposal (algebraic solutions, rigidity, moduli of elliptic curves, mixed families) lie at the interface of differential equations, complex analysis, and algebraic geometry. While they are of independent interest, at least half of them are related to Hilbert's problem. The theory of dynamical systems is divided into two parts: multidimensional systems (the realm of chaos) and two-dimensional systems (the realm of order). Hilbert's sixteenth problem is a central one in the theory of two-dimensional systems. It is well known that two-dimensional dynamical systems provide models for various problems in physics, engineering, and biology (e.g., predator-prey models in biology). The understanding of real two-dimensional dynamics is therefore a subject of general scientific interest. On the other hand, the study of complex extensions of real dynamical systems provides important new information about real systems and is interesting in its own right. Indeed, some experts even say that the "Book of Nature" is written in the language of complex analysis.