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


Peter Borwein,
Department of Mathematics , Simon Fraser University , 8888 University Drive , Burnaby, British Columbia V5A 1S6
Stephen Choi,
Department of Mathematics , Simon Fraser University , 8888 University Drive , Burnaby, British Columbia V5A 1S6
Ron Ferguson,
Department of Mathematics , Simon Fraser University , 8888 University Drive , Burnaby, British Columbia V5A 1S6
Jonas Jankauskas,
Department of Mathematics and Informatics , Vilnius University , Naugarduko 24, Vilnius LT03225 , Lithuania
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 $k$, $0 \leq k \leq n1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.