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
