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

The class B of lacunary polynomials f(x):=-1+x+xn+xm1+xm2+⋯+xms, where s⩾0, m1-n⩾n-1, mq+1-mq⩾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 -π/18⩽argz⩽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).