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Conjugate Reciprocal Polynomials with All Roots on the Unit Circle

  • Kathleen L. Petersen (a1) and Christopher D. Sinclair (a2)
Abstract

We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree N is naturally associated to a subset of ℝ N−1. We calculate the volume of this set, prove the set is homeomorphic to the N − 1 ball and that its isometry group is isomorphic to the dihedral group of order 2N.

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References
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[6] Farmer, D.W., Mezzadri, F., and Snaith, N. C., Random polynomials, random matrices, and L-functions. II. Nonlinearity 19(2006), no. 4, 919936.
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[9] Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3(1971), 169175.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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