On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials

Abstract

The class B of lacunary polynomials f (x) := −1general theorem of factorization of the almost Newman polynomials of the class B is obtained. Such polynomials possess lenticular roots in the open unit disk off the unit circle in the small angular sector −π/18 arg z π/18 and their nonreciprocal parts are always irreducible. The existence of lenticuli of roots is a peculiarity of the class B . By comparison with the Odlyzko-Poonen Conjecture and its variant Conjecture, an Asymptotic Reducibility Conjecture is formulated aiming at establishing the proportion of irreducible polynomials in this class.This proportion is conjectured to be 3/4 and estimated using Monte-Carlo methods. The numerical approximate value ≈ 0.756 is obtained. The results extend those on trinomials (Selmer) and quadrinomials (Ljunggren, Mills, Finch and Jones).

Alphabets, rewriting trails and periodic representations in algebraic bases

“…The sequence (θ −1 n ) n≥2 is decreasing, tends to 1 if n tends to +∞. Theorem 3.3 (Dutykh -Verger-Gaugry [5]). For any f ∈ B n , n ≥ 3, denote by…”

“…Let us briefly recall what is a lenticular zero of f β . Many examples of lenticular zeroes are given in [5]. The following theorem is Theorem 4 in [5].…”

“…Newman polynomials are polynomials with coefficients in {0, 1}. In [5] almost Newman polynomials have been introduced: an almost Newman polynomial is an integer polynomial which has coefficients in {0, 1} except the constant term equal to −1.…”

“…The "Asymptotic Reducibility Conjecture", formulated in [5], says that 75% of the polynomials f (x) ∈ B are irreducible. The factorization and the zeroes of the polynomials of the class B n , n ≥ 2, have been studied in [5]. Theorem 3.2 (Selmer [24]).…”

“…Let us go back to the above assumption. If Ω is a lenticular zero of f β (x), then, by [5], all the polynomial sections S * γ s (x) do have also a (unique) lenticular zero close to Ω which is a conjugate of γ −1 s . For the non-lenticular zeroes of f β (x), very close to the unit circle, the above assumption is necessary.…”