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Unimodular Roots of\\ Special Littlewood Polynomials

  Published:2006-09-01
 Printed: Sep 2006
  • Idris David Mercer
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Abstract

We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if $n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.
MSC Classifications: 26C10, 30C15, 42A05 show english descriptions Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
Trigonometric polynomials, inequalities, extremal problems
26C10 - Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
30C15 - Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
42A05 - Trigonometric polynomials, inequalities, extremal problems
 

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