Cyclotomic polynomials and minimal sets of Lefschetz periods

Abstract

Let qðtÞ ¼ c n1 ðtÞ· · ·c nk ðtÞ, where c ni ðtÞ is the n i -th cyclotomic polynomial. Let z q ðtÞ ¼ qðtÞð1 2 tÞ 22 or z q ðtÞ ¼ qðtÞð1 2 t 2 Þ 21 , depending if the leading coefficient of the polynomial qðtÞ is ' þ 1' or ' 2 1', respectively. The rational function z q ðtÞ can be written as, the r i 's are positive integers, m i 's are integers and N z is a positive integer depending on z q . In the present paper, we study the set L z :¼ >{r 1 ; . . . ; r Nz } where the intersection is considered over all the possible decompositions of z q ðtÞ of the type mentioned above. Here, we describe the set L z 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 z q ðtÞ, such that L z ¼ S? The set L z 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.

Minimal sets of periods for Morse–Smale diffeomorphisms on non-orientable compact surfaces without boundary

“…In [17] it was shown that for all orientable compact surfaces without boundary there are C The results of statements (f) and (g) of Theorem 9 also hold for C 1 Morse-Smale diffeomorphisms on orientable compact surfaces without boundary, and their proof are similar, see [17].…”

“…In [17] this open question was stated as a conjecture for the C 1 Morse-Smale diffeomorphisms on orientable compact surface without boundary.…”