Polynomials with Nonnegative Coefficients
نویسندگان
چکیده
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.
منابع مشابه
Unimodality and Log-Concavity of Polynomials
A polynomial is unimodal if its sequence of coefficients are increasing up to an index, and then are decreasing after that index. A polynomial is logconcave if the sequence of the logarithms of the coefficients is concave. We prove that if P (x) is a polynomial with nonnegative non-decreasing coefficients then P (x+z) is unimodal for any natural z. Furthermore, we prove that if P (x) is a log-c...
متن کاملGale duality bounds for roots of polynomials with nonnegative coefficients
We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. We apply our technique to bound the loc...
متن کاملA Necessary and Sufficient Condition for Nonnegative Product Linearization of Orthogonal Polynomials
cos nθ cos mθ = 2 cos(n − m)θ + 2 cos(n + m)θ. Certain classical orthogonal polynomials admit explicit computation of the coefficients c(n,m, k). For example, they are known explicitly for the ultraspherical polynomials along with their q-analogs [8]. However, they are not available in a simple form for the nonsymmetric Jacobi polynomials (see [7]). The first general criterion for nonnegativity...
متن کاملJack polynomials and free cumulants
We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group. 2000 Mathematics Subject Classification: 05...
متن کاملOn the Product of Log-concave Polynomials
A real polynomial is called log-concave if its coefficients form a log-concave sequence. We give a new elementary proof of the fact that a product of log-concave polynomials with nonnegative coefficients and no internal zero coefficients is again log-concave. In addition, we show that if the coefficients of the polynomial ∏ m∈M(x + m) form a monotone sequence where M is a finite multiset of pos...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999