Cyclotomic polynomials and minimal sets of Lefschetz periods

Let q(t) = c(n1)(t)...c(nk)(t), where c(ni)(t) is the n(i)-th cyclotomic polynomial. Let zeta(q)(t) = q(t)(1 - t)(-2) or zeta(q)(t) = q(t)(1 - t(2))(-1), depending if the leading coefficient of the polynomial q(t) is ' + 1' or ' - 1', respectively. The rational function zeta(q)(t) can be written as Pi(N zeta)(i=1)(1 + Delta(i)t(ri))(mi), where Delta(i) = +/- 1, the r(i)'s are positive integers, m(i)'s are integers and N-zeta is a positive integer depending on zeta(q). In the present paper, we study the set L-zeta := boolean AND{r(1), ... , r(N zeta)} where the intersection is considered over all the possible decompositions of zeta(q)(t) of the type mentioned above. Here, we describe the set L-zeta in terms of the arithmetic properties of the integers n(1), ... , n(k). We also study the question: given S a finite subset of the natural numbers, does exists a zeta(q)(t), such that L-zeta = S? The set L-zeta 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).