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

A generalization of the Bombieri-Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's 'Theorem 14' by real-analytic considerations alone.Comment: 13 page

Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation

International audienceWe consider an Abel differential equation y′=p(x)y2+q(x)y3 with p(x), q(x) polynomials in x. For two given points a and b in C, the ``Poincar\'e mapping" of the above equation transforms the values of its solutions at a into their values at b. In this paper we study global analytic properties of the Poincar\'e mapping, in particular, its analytic continuation, its singularities and its fixed points (which correspond to the ``periodic solutions" such that y(a)=y(b)). On the one hand, we give a general description of singularities of the Poincar\'e mapping, and of its analytic continuation. On the other hand, we study in detail the structure of the Poincar\'e mapping for a local model near a simple fixed singularity, where an explicit solution can be written. Yet, the global analytic structure (in particular, the ramification) of the solutions and of the Poincar\'e mapping in this case is fairly complicated, and, in our view, highly instructive. For a given degree of the coefficients we produce examples with an infinite number of {\it complex} ``periodic solutions" and analyze their mutual position and branching. Let us recall that Pugh's problem, which is closely related to the classical Hilbert's 16th problem, asks for the existence of a bound to the number of {\it real} isolated ``periodic solutions". New findings reported here lead us to propose new insights on the Poincar\'e mapping. If the ``complexity" of the path in the x-plane between a and b is {\it a priori} bounded, the number of fixed points should be uniformly bounded. We think that, in some sense, this is close to the complex version of Khovansky's fewnomial theory

Linear versus Non-linear Acquisition of Step-Functions

We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and nonlinear approaches. As a model case we use piecewise-constant functions on [0, 1], in particular, the Heaviside jump function Ht = χ [0,t]. Considered as a curve in the Hilbert space L 2 ([0, 1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov’s n-width and ɛ-entropy, as well as Temlyakov’s (N, m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other. This research was supported by the ISF, Grant No. 304/05, and b