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Elementary Symmetric Polynomials in Numbers of Modulus 1

Published online by Cambridge University Press:  20 November 2018

Donald I. Cartwright  and
Tim Steger
Donald I. Cartwright
Affiliation:
School of Mathematics and Statistics University of Sydney N.S.W. 2006 Australia
Tim Steger
Affiliation:
Istituto di Matematica e Fisica Università di Sassari via Vienna 2 07100 Sassari Italy
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Abstract

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We describe the set of numbers ${{\sigma }_{k}}\left( {{z}_{1}},\cdot \cdot \cdot ,{{z}_{n+1}} \right)$, where ${{z}_{1}},\cdot \cdot \cdot ,{{z}_{n+1}}$ are complex numbers of modulus 1 for which ${{z}_{1}}{{z}_{2}}\cdot \cdot \cdot {{z}_{n+1}}=1$, and ${{\sigma }_{k}}$ denotes the $k$-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type ${{\overset{\sim }{\mathop{\text{A}}}\,}_{n}}$.

Keywords

05E0533C4530C1551E24
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 2 , 01 April 2002 , pp. 239 - 262
Copyright
Copyright © Canadian Mathematical Society 2002

References

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