Positivity conditions for polynomials

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.

Asymptotic expansions and positivity of coefficients for largepowers of analytic functions

“…In fact, it is shown in [3,Theorem 6.4] that if f is strongly positive on a neighborhood of r , then σ > 0 (that theorem was stated and proved for polynomials, but the same proof can be used verbatim for power series).…”

“…It can be shown that if f is the pointwise spectral radius of an irreducible matrix whose entries are polynomials with positive integral coefficients (the beta function of Tuncel, see [10]), then µ f and σ f are closely related to the entropy and information variance of the associated Markov chain. The function µ f is the Legendre transformation of f (see, e.g., [3]). …”

“…We also say that f is numerically positive if f (r ) > 0 for r > 0. Condition (1.4) arises in the study of finite state Markov chains, and is discussed in [3]. A simple criterion to check (1.4) for polynomial f at all sufficiently small and sufficiently large r is given in [4].…”

Characterization of polynomials whose large powers have all positive coefficients

“…Positivity conditions for polynomials with real coefficients play a key role in several branches of mathematics, such as real algebraic geometry, convex geometry, probability theory and optimization, and have been widely studied (see e.g. [3,9,11,15,16,17,18,19,20,23,24] and the references therein). An interesting and important class of polynomials are those whose coefficients are positive.…”

Estimating the kth coefficient of (f(z))n when k is not too large

“…It is proved in [1,Theorem 6.4] that strong positivity on an open interval containing r is enough to ensure that σ f (r) > 0. Hence for strongly positive polynomials, μ f provides a bijection from (0, ∞) to (0, deg f ).…”