In 2002, Dierk Schleicher gave an explicit estimate of an upper bound for the number of iterations of Newton's method it takes to find all roots of polynomials with prescribed precision. In this paper, we provide a method to improve the upper bound given by D. Schleicher. We give here an iterative method for finding an upper bound for the distance between a fixed pointzin an immediate basin of a rootαtoα, which leads to a better upper bound for the number of iterations of Newton's method.

Abstract. Let D denote the unit disk {z : |z| < 1} in the complex plane C. In this paper, we study a family of polynomials P with only one zero lying outside D. We establish criteria for P to satisfy implying that each of P and P has exactly one critical point outside D.1. Introduction. Let P be a polynomial in the complex plane C. We denote the degree of P by deg P . We say that α is a critical point of P if P (α) = 0. Throughout this paper, if not otherwise stated, when we talk about the number of zeros of a polynomial in a domain, we mean the number of zeros counting multiplicities. As the critical points of P are the zeros of P , this applies also to the number of critical points. There are several known results involving the critical points of polynomials. The most classical one is the Gauss-Lucas Theorem, [8, p. 25].

In this paper, we study affine mappings on domains in R 3 that are unions of simplices. Let f i be an affine mapping of the form f i (x) = A i x, where A i is a 3×3 transformation matrix, on a simplex Q i . We establish the condition of these matrices A i in order to obtain a continuous piecewise affine mapping f on the domain Q that is the union of simplices Q i such that f |Q i = f i .

In this paper, we establish a necessary and sufficient condition for two algebraic integers in complex quadratic number fields to be consecutive terms of generalized Fibonacci numbers. We use this result to obtain all solutions of the Diophantine equation x 2 − a x y + b y 2 = c over Gaussian integers, where b and c are units in [i] and a ∈ [i] with |a| 2 10.