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 $p\left( z \right)$ (polynomials with coefficients {−1, 1}) inside or on the unit circle $\left| z \right|\, =\, 1$, denoted by $N\left( p \right)$ and $U\left( p \right)$, 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 $N\left( p \right)$, $U\left( p \right)$ for polynomials $p\left( z \right)$ 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 $p\left( z \right)$ of degree $n$ with $N\left( p \right)\, =\, k$ and $U\left( p \right)\, =\, 0$ . Furthermore, we describe some cases where the ratios $N\left( p \right)/n$ and $U\left( p \right)/n$ have limits as $n\, → \, ∞ $ and find the corresponding limit values.