On Properties Of Some Special Polynomials Over Arithmetic Fields

The polynomials over a field are one of the basic objects in algebra.The properties of polynomials over special fields,such as finite fields and local fields,take on a critical role in sequential cryptography and coding.In this article,we study the periodic properties of a kind of polynomials over finite fields and the related dynamical properties of morphisms on higher dimensional projective space over local field.Let σ（x）=x^q2+α₁x^q+α₂x∈F_q[x].In this article,we first study the periodic properties of the iteration of σ（x）over finite field F_q.When the coefficient α₁,α₂ of σ（x）and positive integer m satisfy certain conditions,we study properties of iterative polynomialσ^m（x）-x and the properties of periodic points with period n.Secondly,inspired by Kawaguchi’s work on morphisms on higher dimensional projective space and canonical height,we study dynamical properties of morphisms on higher dimensional projective space over local field.This article is framed as follows:In part one,we introduce the significance and background of the article.The second part mainly explains the basic knowledge of discrete dynamical systems,the iterates of polynomials over finite fields,and local field and the associated definitions and properties.In chapter three,we investigate the periodic properties of iteration of a special linear polynomialσ（x）∈F_q[x].When q and m satisfy certain conditions,and the order of the coefficientα₂ of σ（x）divides m,we obtain that σ^m（x）-x is a q-th powers.Otherwise,σ^m（x）-x has no multiple factors.In the fourth chapter,we study the dynamical properties of morphisms on higher dimensional projective space over local field.In this chapter,we study the properties of reduction of points in high dimensional projective space and the linear fraction reduction by the chordal metric.By using the Green function g_Φ of Φ,we introduce the arithmetic distance of morphisms and investigate its property.The necessary and sufficient condition which Φ has good reduction is obtained in this paper.We also describe explicitly the Filled Julia set of Φ by its Green function.The last chapter summarizes and prospects the text.