Abstract view
Unimodular Roots of\\ Special Littlewood Polynomials


Published:20060901
Printed: Sep 2006
Abstract
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n1} z^{n1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ selfreciprocal
if $\alpha(z) = z^{n1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (1)^j a_{mj}$ 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 selfreciprocal
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
