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Polynomial beta functions

Published online by Cambridge University Press:  19 September 2008

Valerio De Angelis
Valerio De Angelis
Affiliation:
Department of Mathematics, University of Ottawa, 585 King Edward, Ottawa, Ontario KIN 6N5, Canada
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Abstract

The pointwise spectral radii of irreducible matrices whose entries are polynomials with positive, integral coefficients are studied in this paper. Most results are derived in the case that the resulting algebraic function, the beta function of S. Tuncel, is in fact a polynomial. We show that the set of beta functions forms a semiring, and the spectral radius of a matrix of beta functions is again a beta function. We also show that the coefficients of a polynomial beta function p must be real algebraic integers, and p satisfies (after a change of variables if necessary) the inequality for non-zero (and not all positive) complex numbers z1,…,zd. If and the ordered sequence of exponents appearing in p is of the form (m,m+1,…,M−,1,M) for some integers m and M, the same inequality is necessary and sufficient for p to be a beta function.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 3 , September 1994 , pp. 453 - 474
Copyright
Copyright © Cambridge University Press 1994

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