Divisibility At Integer Points Of Multivariate Polynomials And Indefinite Equations On PID

Integral coefficient polynomials which is a major research problem of algebraic number theory and algebraic geometry are important polynomials.Indefinite linear equations are an important class of algebraic equations,which are widely used in many scientific and technological problems.This article focus on the divisibility of multiple integral coefficient polynomials at whole points and the existence of the solution of the indefinite quasilinear equations.This paper is divided into four chapters:Chapter 1 briefly introduces the research background and main work of this paper.Chapter 2 mainly introduces the basic knowledge of number theory and algebra related to this paper.Chapter 3 discusses the divisibility of the multivariate integral polynomial at the whole points.It includes three special types of multivariate integral coefficient polynomials.through the remainder division method and the reversible transformation,we can get Conditions of divisibility at the whole point.For situations which cannot be discussed on the integer ring,we try to discuss this problem on Gaussian rings and then came to a conclusion.Chapter 4 discusses the existence of solutions of indefinite quasilinear equations on the principal ideal integral ring PID.By using the properties of the principal ideal integral ring,the existence of the solution of the indefinite linear equations(4.1)is obtained by the elementary transformation of the matrix.Add an squared term to the equations(4.1).On an existing basis,we use similar method to study the existence of the solution of the indefinite quasilinear equations.