On Some Bounds for Zeros of Norm-Bounded Polynomials

We define a region Hα,ƒ in the complex number field, where α is a complex number, ƒ(x) ε K [x] and ƒ(α) ≠ 0. The region Hα,ƒ contains no zeros of ƒ(x) and is relatively easy to analyze. We analyze the region with respect to K = and K= . By the results of the analysis, we derived some bounds for zeros of ƒ(x) from the norm of ƒ(x). The region Hα,ƒ can be used for the analysis of the distribution of zeros of polynomials over integers whose norms and degrees are bounded. For these polynomials, we calculated the distributions of their zeros by computer and compared them with the regions. For several cases the regions describe the distributions well. However, there are some cases where the regions do not describe well.