TAILIEUCHUNG - Báo cáo toán học: "On the Modes of Polynomials Derived from Nondecreasing Sequences"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On the Modes of Polynomials Derived from Nondecreasing Sequences. | On the Modes of Polynomials Derived from Nondecreasing Sequences Donna Q. J. Dou School of Mathematics Jilin University Changchun 130012 P. R. China qj dou@ Arthur L. B. Yang Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 P. R. China yang@ Submitted Oct 13 2010 Accepted Dec 15 2010 Published Jan 5 2011 Mathematics Subject Classification 05A20 33F10 Abstract Wang and Yeh proved that if P x is a polynomial with nonnegative and nondecreasing coefficients then P x d is unimodal for any d 0. A mode of a unimodal polynomial f x a0 a1x amxm is an index k such that ak is the maximum coefficient. Suppose that M P d is the smallest mode of P x d and M P d the greatest mode. Wang and Yeh conjectured that if d2 di 0 then M P d1 M P d2 and M P d1 M P d2 . We give a proof of this conjecture. Keywords unimodal polynomials the smallest mode the greatest mode. 1 Introduction This paper is concerned with the modes of unimodal polynomials constructed from nonnegative and nondecreasing sequences. Recall that a sequence a-. o - m is unimodal if there exists an index 0 k m such that a0 ak-1 ak ak 1 am- Such an index k is called a mode of the sequence. Note that a mode of a sequence may not be unique. The sequence ai 0 i m is said to be spiral if am ao Om-1 ai a m THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P1 1 where m stands for the largest integer not exceeding 22. Clearly the spiral property implies unimodality. We say that a sequence ai 0 i m is log-concave if for 1 k m 1 2 n .n ak ak 1ak-1 and it is ratio monotone if am a0 Zm- a1 ai am- m-1 a 1 and a0 am- a1 1 am-2 ai-1 am-i a m -1 - am- m 1. It is easily checked that ratio monotonicity implies both log-concavity and the spiral property. Let P x a0 a1x amxm be a polynomial with nonnegative coefficients. We say that P x is unimodal if the sequence ai 0 i m is unimodal. A mode of ai 0 i m is also called a mode of P x . Similarly we say that P x is log-concave or ratio .

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