Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-07T05:20:06.230Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugate Reciprocal Polynomials with All Roots on the Unit Circle

Published online by Cambridge University Press:  20 November 2018

Kathleen L. Petersen  and
Christopher D. Sinclair
Kathleen L. Petersen
Affiliation:
Queen’s University, Kingston, ON, K7l 3N5 e-mail:[email protected]
Christopher D. Sinclair
Affiliation:
Pacific Institute for the Mathematical Sciences, Vancouver, BC, V6T 1Z4 e-mail:[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 ${{\mathbb{R}}^{N-1}}$. 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 $2N$.

Keywords

11C0828A7515A5254H1058D19
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 5 , October 2008 , pp. 1149 - 1167
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Bogomolny, E., Bohigas, O., and Leboeuf, P., Quantum chaotic dynamics and random polynomials. J. Statist. Phys. 85(1996), no. 5-6, 639–679.Google Scholar
[2] Bonsall, F. F. and Marden, M., Zeros of self-inversive polynomials. Proc. Amer. Math. Soc. 3(1952), 471–475.Google Scholar
[3] Chern, S.-J. and Vaaler, J. D., The distribution of values of Mahler's measure. J. Reine Angew. Math. 540(2001), 1–47.Google Scholar
[4] DiPippo, S. A. and Howe, E.W., Real polynomials with all roots on the unit circle and abelian varieties over finite fields. J. Number Theory 73(1998), no. 2, 426–450.Google Scholar
[5] Dyson, F. J., Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19(1970), 235–250.Google Scholar
[6] Farmer, D.W., Mezzadri, F., and Snaith, N. C., Random polynomials, random matrices, and L-functions. II. Nonlinearity 19(2006), no. 4, 919–936.Google Scholar
[7] Schinzel, A., Self-inversive polynomials with all zeros on the unit circle. Ramanujan J. 9(2005), no. 1-2, 19–23.Google Scholar
[8] Sinclair, C. D., The range of multiplicative functions on ℂ[x], ℝ[x] and ℤ[x]. Proc. London Math. Soc. 96(2008), no. 3, 697–737.Google Scholar
[9] Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc. 3(1971), 169–175.Google Scholar