Polynomials with nonnegative coefficients

Abstract

The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen’s inequality. The conjecture was also made independently by B. Conroy in connection with his work in number theory. A crucial and interesting lemma is proved which describes general coefficient-root relations for polynomials with nonnegative coefficients and for polynomials for which the case of equality holds in Descarte’s Rule of Signs.

Adaptive Polling in Hierarchical Social Networks Using Blackwell Dominance

“…It is intuitive that the opinion fractions in (27) from nodes at level i are more informative than opinion fractions from nodes at level j(> i) in Fig.1 owing to obvious Blackwell dominance relation of opinion distributions B l for l = i, j in (10).…”

“…Here N j and n (j) i indicate the total and the number in favor of x = i reported and B l denotes the opinion distribution (10) at level l. The likelihood in (27) is the well known multinomial distribution. Remark: In case of adaptive friendship polling, the nodes report opinion fractions to the pollster.…”

“…Information Theoretic Consequence: Let I(X ; Y (u) ) denote the mutual information of channel O(u) and C (u) denote the capacity defined in (4). Let (O(u)) i denote the i th row of the multinomial likelihood O(u) in (27). ii.)…”

On the Lorentz degree of a product of polynomials

“…We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al (1991) on polynomials with nonnegative coefficients. …”

“…In Section 3, we establish a link between the Lorentz degree and the Pólya degree of polynomials. This allows us to use the results of Barnard et al [1] to prove the first inequality (1) for degree 2 polynomials and a family of polynomials of higher degree. Counter-examples to the two inequalities (1) and (2) are given in Section 4.…”