On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk. (1st June 2015) - Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas
Abstract: We investigate the numbers of complex zeros of Littlewood polynomials p( z ) (polynomials with coefficients −1, 1) inside or on the unit circle | z | = 1, denoted by N( p ) and U( p ), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for N( p ), U( p ) for polynomials p( z ) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 ⩽ k ⩽ n-1, there exists a Littlewood polynomial p( z ) of degree n with N( p ) = k and U( p ) = 0 . Furthermore, we describe some cases where the ratios N( p )/n and U( p )/n have limits as n → ∞ and find the corresponding limit values.