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Elementary Symmetric Polynomials in Numbers of Modulus 1

  • Donald I. Cartwright (a1) and Tim Steger (a2)
Abstract

We describe the set of numbers σk (z 1,…,z n+1), where z 1,…,z n+1 are complex numbers of modulus 1 for which z 1 z 2z n+1 = 1, and σ 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 à n.

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References
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[Ca1] Cartwright, D. I., Harmonic functions on buildings of type Ãn. In: Random Walks and Discrete Potential Theory, (eds., M. Picardello and W. Woess), Symposia Math. XXXIX, Cambridge Univ. Press, 1999, 104138.
[Ca2] Cartwright, D. I., Spherical harmonic analysis on buildings of type Ãn. Monatsh. Math., to appear.
[CM] Cartwright, D. I. and Młotkowski, W., Harmonic analysis for groups acting on triangle buildings. J. Austral. Math. 56 (1994), 345383.
[Ll] Lloyd, N. G., Degree Theory. Cambridge Tracts in Math. 73, Cambridge Univ. Press, 1978.
[Mac1] Macdonald, I. G., Spherical functions on a group of p-adic type. Ramanujan Inst. Publications 2 , University of Madras, 1971.
[Mac2] Macdonald, I. G., Symmetric functions and Hall polynomials, second edition. Oxford Univ. Press, 1995.
[MZ] Mantero, A. M. and Zappa, A., Spherical functions and spectrum of the Laplace operators on buildings of rank 2. Boll. Un. Mat. Ital. B (6) 8 (1994), 419475.
[Ro] Ronan, M., Lectures on buildings. Academic Press, 1989.
[Sz] Szegö, G., Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. XXIII, Amer. Math. Soc., Providence, Rhode Island, 1939.
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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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