Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R...Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11071030,11371078)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20110041110039)
文摘Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
