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

TL;DR

This paper investigates the factorization and root distribution of a special class of lacunary polynomials called almost Newman polynomials, revealing lenticular roots and proposing an asymptotic conjecture on their irreducibility proportion.

Contribution

It provides a general factorization theorem for almost Newman polynomials and introduces the asymptotic reducibility conjecture with numerical support.

Findings

Lenticular roots are found in a small angular sector off the unit circle.

Nonreciprocal parts of these polynomials are always irreducible.

Estimated irreducibility proportion is approximately 0.756, close to the conjectured 3/4.

Abstract

The class B of lacunary polynomials f(x) := -1 + x + x^n + x^{m_1} + x^{m_2} + ... + x^{m_s}, where s >= 0, m_1 - n >= n - 1, m_{q+1} - m_{q} >= n - 1 for 1 <= q < s, n >= 3 is studied. A polynomial having its coefficients in {0, 1} except its constant coefficient equal to -1 is called an almost Newman polynomial. A general 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 \pi/18 <= arg z <= \pi/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).