Determining the limits of bivariate rational functions by Sturm's theorem

Abstract

In this paper, we present an algorithm for determining the limits of real rational functions in two variables, based on Sturm's familiar theorem and the general Sturm–Tarski theorem for counting certain roots of univariate polynomials in a real closed field. Let R[x,y] be the ring of polynomials with real coefficients in two variables x, y, and let u(x,y), v(x,y)∈R[x,y] be two non-zero polynomials such that u(a,b)=v(a,b)=0 for a, b∈R. The purpose of this paper is to decide the existence of lim(x,y)→(a,b)⁡u(x,y)v(x,y) and compute the limit if it exists. Our algorithm needs no assumption on the denominators and does not involve the computation of Puiseux series.