Cyclotomic polynomials and minimal sets of Lefschetz periods

Abstract

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the 's are positive integers, 's are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855).