On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk

Borwein, Peter ; Choi, Stephen ; Ferguson, Ron ; Jankauskas, Jonas

Canadian journal of mathematics, Tome 67 (2015) no. 3, pp. 507-526

Résumé

We investigate the numbers of complex zeros of Littlewood polynomials $p\left( z \right)$ (polynomialswith 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 inthe sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtainexplicit 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\,\to \,\infty $ and find the corresponding limit values.

Détail

DOI : 10.4153/CJM-2014-007-1

Mots-clés : 11R06, 11R09, 11C08, Littlewood polynomials, zeros, complex roots