tailieunhanh - Báo cáo toán học: " A criterion for unimodality"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A criterion for unimodality. | A criterion for unimodality George Boros Department of Mathematics University of New Orleans New Orleans LA 70148 gboros@ Victor H. Moll1 Department of Mathematics Tulane University New Orleans LA 70118 vhm@ Submitted January 23 1999 Accepted February 2 1999 Classification 05 33 40 Abstract We show that if P x is a polynomial with nondecreasing nonnegative coefficients then the coefficient sequence of P x 1 is unimodal. Applications are given. 1. Introduction A finite sequence of real numbers do d1 dm 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. Unimodal polynomials arise often in combinatorics geometry and algebra. The reader is referred to 2 and 3 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 logarithmically concave or log-concave for short if dj 1 dj-1 dj for 1 j m 1. It is easy to see that if a sequence is log-concave then it is unimodal 4 . 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 4 . A second criterion for the log-concavity of a polynomial was determined by Brenti 2 . A sequence of real numbers is said to have no internal zeros if whenever difdk 0 and i j k then dj 0. Brenti s criterion states that if P x is a log-concave polynomial with nonnegative coefficients and with no internal zeros then P x 1 is log-concave. 1 www http 80 vhm THE ELECTRONIC JOURNAL OF COMBINATORICS 6 1999 R10 2 2. The main result Theorem . If P x is a polynomial with positive nondecreasing coefficients then P x 1 is unimodal. Proof. Observe first that Pm