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# Conjugate Reciprocal Polynomials with All Roots on the Unit Circle

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Published:2008-10-01
Printed: Oct 2008
• Kathleen L. Petersen
• Christopher D. Sinclair
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## 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 $2N$.
 MSC Classifications: 11C08 - Polynomials [See also 13F20] 28A75 - Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 15A52 - Random matrices54H10 - Topological representations of algebraic systems [See also 22-XX] 58D19 - Group actions and symmetry properties

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