tailieunhanh - Báo cáo toán học: "An extension of a criterion for unimodality"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: An extension of a criterion for unimodality. | An extension of a criterion for unimodality Jenny Alvarez Department of Mathematics UC Santa Barbara CA USA jalvar01@ Miguel Amadis Department of Mathematics Nyack College New York NY USA amadism@ George Boros Department of Mathematics Xavier University New Orleans LA 70125 USA gboros@ Dagan Karp Department of Mathematics Tulane University New Orleans LA 70118 USA dkarp@ Victor H. Moll Leobardo Rosales Department of Mathematics Department of Mathematics Tulane University New Orleans LA 70118 USA UC San Diego CA USA vhm@ lrosales@ Submitted March 20 2001 Accepted September 19 2001. Subject Classifications 40 33 05 Abstract We prove that if P x is a polynomial with nonnegative nondecreasing coefficients and n is a positive integer then P x n is unimodal. Applications and open problems are presented. 1 Introduction A finite sequence of real numbers d0 d1 dmg is said to be unimodal if there exists an index 0 m m called the mode of the sequence such that dj increases up to j m and decreases from then on that is do d1 dm and dm dm 1 dm. A polynomial is said to be unimodal if its sequence of coefficients is unimodal. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R30 1 Unimodal polynomials arise often in combinatorics geometry and algebra. The reader is referred to 3 and 4 for surveys of the diverse techniques employed to prove that specific families of polynomials are unimodal. A sequence of positive real numbers d0 d1 dm is said to be logarithmic concave or log concave for short if dj 1dj_1 dj for 1 j m 1. It is easy to see that if a sequence is log concave then it is unimodal 5 . A sufficient condition for log concavity of a polynomial is given by the location of its zeros if all the zeros of a polynomial are real and negative then it is log concave and therefore unimodal 5 . A simple criterion for unimodality was established in 2 if aj is a nondecreasing sequence of positive real numbers then m P x 1