Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T04:30:39.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Positivity conditions for polynomials

Published online by Cambridge University Press:  19 September 2008

Valerio De Angelis
Valerio De Angelis
Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, USA‡
Get access Rights & Permissions [Opens in a new window]

Abstract

We study several notions of positivity for a class of real-valued functions of several variables that includes the Laurent polynomials. We show that Handelman's positivity condition is characterized by boundedness of the associated Legendre transformation, or boundedness of the entropy function, while another notion of positivity here introduced characterizes the mapping property of the Legendre transformation derived by Marcus and Tuncel for beta functions. Several examples are given to distinguish the various notions of positivity.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 1 , March 1994 , pp. 23 - 51
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]De Angelis, V.. PhD dissertation. University of Washington, 1992.Google Scholar
[2]De Angelis, V.. Coefficients of large powers of a polynomial. Preprint.Google Scholar
[3]De Angelis, V.. Polynomial beta functions. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[4]Goodearl, K. & Handelman, D.. Partially ordered Abelian groups with interpolation. Amer. Math. Soc. Mathematical Surveys and Monographs. 20 (1986).Google Scholar
[5]Grünbaum, B.. Convex polytopes. Pure and Applied Mathematics. Vol. 16. Interscience Publishers, New York, 1967.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 Lecture Notes in Mathematics. 1282 Springer, Berlin, 1987.Google Scholar
[8]Handelman, D.. Polynomials with a positive power. Contemp. Math. 135 (1992), 229–230.CrossRefGoogle Scholar
[9]Marcus, B. & Tuncel, S.. The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains. Ergod. Th. & Dynam. Sys. II (1991), 129–180.CrossRefGoogle Scholar
[10]Marcus, B. & Tuncel, S.. Entropy at a weight-per-symbol and embeddings of Markov chains. Invent. Math. 102 (1990), 235–266.CrossRefGoogle Scholar
[11]Odlyzko, A. M. & Richmond, L. B.. On the unimodality of high convoluions of discrete distributions. The Annals of Probability. 13 (1985), 299–306.CrossRefGoogle Scholar
[12]Parry, W. & Tuncel, S.. Classification problems in Ergodic Theory. LMS Lecture Notes 67 Cambridge University Press: Cambridge, 1982.Google Scholar
[13]Parry, W. & Tuncel, S.. On the stochastic and topological structure of Markov chains. Bull. Lond. Math. Soc. 14 (1982), 16–27.CrossRefGoogle Scholar
[14]Tuncel, S.. Faces of Markov chains and matrices of polynomials. Contemp. Math. 135 (1992), 391–422CrossRefGoogle Scholar