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Unimodular Roots of Special Littlewood Polynomials

  • Idris David Mercer (a1)
Abstract

We call α(z) = a 0 + a 1 z + · · · + an –1 zn –1 a Littlewood polynomial if aj = ±1 for all j. We call α(z) self-reciprocal if α(z) = zn –1α(1/z), and call α(z) skewsymmetric if n = 2m + 1 and am +j = (–1) j am j for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in ℂ 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.

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References
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[1] Borwein, P. B., Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics 10, Springer-Verlag, New York (2002).
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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