On a Property of Roots of Polynomials

Abstract:

      In [1], 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 polynomial λ_nzⁿ+…+λ₂z²+λ₁z+λ₀ of degree n has a root α such that inf{｜λ_nα^nm+…+λ₂a^2m+λ₁α^m+λ₀｜：m=2，3，…｝>0 if and only if at least one of the following two conditions holds: (i) the polynomial has a root βsatisfying ｜β｜ ＞ 1; (ii) the polynomial has a root βsatisfying ｜β｜ < 1, and λ₀ ≠ 0.