Determination of the limits for multivariate rational functions

Abstract

The purpose of this paper is to solve the problem of determining the limits of multivariate rational functions. It is essential to decide whether or not {ie1-1} for two non-zero polynomials f, g ∈ ℝ[x ₁, …, x _n ] with f(0, …, 0) = g(0, …, 0) = 0. For two such polynomials f and g, we establish two necessary and sufficient conditions for the rational function {ie1-2} to have its limit 0 at the origin. Based on these theoretic results, we present an algorithm for deciding whether or not {ie1-3}, where f, g ∈ ℝ[x ₁, …, x _n ] are two non-zero polynomials. The design of our algorithm involves two existing algorithms: one for computing the rational univariate representations of a complete chain of polynomials, another for catching strictly critical points in a real algebraic variety. The two algorithms are based on the well-known Wu’s method. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program. In the final section of this paper, several examples are given to illustrate the effectiveness of our algorithm.