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

摘要

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) epsilon R[x, y] be two nonzero polynomials such that u(a, b) = v(a, b) = 0 for a, b epsilon R. Thepurpose 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. (C) 2019 Elsevier Ltd. All rights reserved.