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On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
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- Journal:
- Canadian Journal of Mathematics / Volume 67 / Issue 3 / 01 June 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 507-526
- Print publication:
- 01 June 2015
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- Article
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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\,\to \,\infty $ and find the corresponding limit values.
