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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 LT-03225 , Lithuania
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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 $k$, $0 \leq k \leq n-1$, there exists a Littlewood polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore, we describe some cases when the ratios $N(p)/n$ and $U(p)/n$ have limits as $n \to \infty$ and find the corresponding limit values.
Keywords: Littlewood polynomials, zeros, complex roots Littlewood polynomials, zeros, complex roots
MSC Classifications: 11R06, 11R09, 11C08 show english descriptions PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Polynomials (irreducibility, etc.)
Polynomials [See also 13F20]
11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R09 - Polynomials (irreducibility, etc.)
11C08 - Polynomials [See also 13F20]
 

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