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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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References

REFERENCES

[1]Angelis, V. De. Polynomial beta functions and positivity of polynomials. PhD dissertation. University of Washington, 1992.Google Scholar
[2]Angelis, V. De. Positivity conditions for polynomials. Ergod. Th. & Dynam. Sys. 14 (1994), 23–51.CrossRefGoogle Scholar
[3]Angelis, V. De. An asymptotic expansion and log concavity of polynomials. Preprint.Google Scholar
[4]Friedland, S.. Limit eigenvalues of non-negative matrices. L. Alg. Appl. (1986), 173–178.CrossRefGoogle Scholar
[5]Gantmacher, F. R.. The Theory of Matrices. Vol. 2. Chelsea: New York, 1959.Google Scholar
[6]Handelman, D.. Positive polynomials and product type actions of compact groups. Memoirs of the Amer. Math. Soc. 320 (1985).Google Scholar
[7]Handelman, D.. Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem. Springer-Verlag Lecture Notes in Mathematics 1282 (1987).CrossRefGoogle Scholar
[8]Handelman, D.. Eventual positivity and finite equivalence for matrices of polynomials. Preprint.Google Scholar
[9]Handelman, D.. Polynomials with a positive power in ‘Symbolic dynamics and its applications’. Contemp. Math. Amer. Math. Soc. 135 (1992), 229–230.CrossRefGoogle Scholar
[10]Handelman, D.. Private communication.Google Scholar
[11]Lind, D.. Matrices of Perron numbers. J. Number Theory 20 (1992), 211–217.CrossRefGoogle Scholar
[12]Lind, D.. The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th & Dynam. Sys. 4 (1984), 283–300.CrossRefGoogle Scholar
[13]Marcus, B. and Tuncel, S.. The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains. Ergod. Th & Dynam. Sys. 11 (1991), 129–180.CrossRefGoogle Scholar
[14]Marcus, B. and Tuncel, S.. Entropy at a weight-per-symbol and embeddings of Markov chains. Invent. Math. 102 (1990), 235–266.CrossRefGoogle Scholar
[15]Marcus, B. and Tuncel, S.. Matrices of polynomials, positivity, and finite equivalence of Markov chains. J. Amer. Math. Soc. 6 No. 1 (1993), 131–147.CrossRefGoogle Scholar
[16]Parry, W. and Tuncel, S.. On the stochastic and topological structure of Markov chains. Bull. Lond. Math. Soc. 14 (1982), 16–27.CrossRefGoogle Scholar
[17]Rockafeller, T.. Convex Analysis. Princeton: Princeton University Press, 1970.CrossRefGoogle Scholar
[18]Seneta, E.. Non-negative Matrices and Markov Chains. Springer: Berlin, Heidelberg, and New York, 1981.CrossRefGoogle Scholar
[19]Tuncel, S.. Conditional pressure and coding. Israel J. Math 39 (1981), 101–112.CrossRefGoogle Scholar
[20]Tuncel, S.. A dimension, dimension modules, and Markov chains. Proc. London Math. Soc. 46 (1983), 100–116.CrossRefGoogle Scholar
[21]Tuncel, S.. Subsystems, Perron numbers, and continuous homomorphisms of Bernoulli shifts. Ergod. Th. & Dynam. Sys. 9 (1989), 561–570.CrossRefGoogle Scholar
[22]Tuncel, S.. Faces of Markov chains and matrices of polynomials in ‘Symbolic dynamics and its applications’. Contemp. Math., Amer. Math. Soc. 135 (1992), 391–422.CrossRefGoogle Scholar
[23]Weiss, E.. Algebraic Number Theory. McGraw-Hill: New York, 1963.Google Scholar