We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could no...

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 o...

It is of frequent occurrence in problems of both pure and applied mathematics that certain values sought may be specified and must be determined as the roots of a tran-scendental equation. In particular, the equation may be of the class in which the unknown is involved only through the medium of exponential or trigonometric functions, with coefficients which are power functions or essentially s...