Real valuations and the limits of multivariate rational functions

Author:

Xiao Shuijing1,Zeng Xiaoning2,Zeng Guangxing1

Affiliation:

1. Department of Mathematics, Nanchang University, Nanchang, Jiangxi Province, 330031, P. R. China

2. Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong Province, 510303, P. R. China

Abstract

The purpose of this paper is to investigate the limits of multivariate rational functions with the aid of the theory of real valuations. The following is one of our main results. For two nonzero polynomials f, g ∈ ℝ[x1,…,xn] and (a1,…,an) ∈ ℝn, the (finite) limit of the rational function [Formula: see text] at (a1,…,an) does not exist if and only if (1) there exists a sequence u1(x),…,un(x) of polynomials over ℝ in one variable x such that ui(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ≠ 0, but [Formula: see text]; or (2) there exist two sequences u1(x),…,un(x) and w1(x),…,wn(x) of polynomials over ℝ in one variable x such that ui(0) = wi(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ⋅ g(w1(x),…,wn(x)) ≠ 0, but [Formula: see text].

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Algebra and Number Theory

Reference10 articles.

1. Algorithms in Real Algebraic Geometry

2. Real Algebraic Geometry

3. On the extension of real places

4. Ring Theory and Algebra II Lecture Notes in Pure and Applied Mathematics;Lam T. Y.,1980

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