摘要
In [l], a property of roots of polynomials is considered, which involves the existence of local analytic solutions of polynomial-like functional iterative equations. In this paper we discuss this property and obtain a succinct condition to decide whether this property holds. Our main result is: A polynomialλnzn+''' + λ2z2 + λlz + λ0 of degree n has a root or such that inf{|λnanm +... + λ2a2m + λ1am+ λ0|: m = 2, 3,.. .} > 0 if and only if at least one of the following two conditions holds: (i) the polynomial has a root β satisfying ｜β｜ > 1; (ii) the polynomial has a root β satisfying ｜β｜ < 1, and λ0≠0
文献［１］在讨论多项式型的函数迭代方程的局部解析解的存在性时涉及到了多项式的根的一个性质．本文给出了判定该性质是否成立的一个简洁的条件，证明了多项式λｎｚｎ＋…＋λ２ｚ２＋λ１ｚ＋λ０有一个根ａ满足ｉｎｆ｛｜λｎａｎｍ＋…＋λ２ａ２ｍ＋λ１ａｍ＋λ０｜：ｍ＝２；３，…｝＞０当且仅当如下两个条件之中至少有一个成立：（１）该多项式有一个根β满足｜β｜＞１；（ｉｉ）该多项式有一个根β满足｜β｜＜１，且λ０≠０．
基金
Supported by the National Natural Science Foundation of China.
