Limits of quotients of polynomial functions in three variables

Abstract

A method for computing limits of quotients of real analytic functions in two variables was developed in [4]. In this article we generalize the results obtained in that paper to the case of quotients q = f(x, y, z)/g(x, y, z) of polynomial functions in three variables with rational coefficients. The main idea consists in examining the behavior of the function q along certain real variety X(q) (the discriminant variety associated to q). The original problem is then solved by reducing to the case of functions of two variables. The inductive step is provided by the key fact that any algebraic curve is birationally equivalent to a plane curve. Our main result is summarized in Theorem 2. In Section 4 we describe an effective method for computing such limits. We provide a high level description of an algorithm that generalizes the one developed in [4], now available in Maple as the limit/multi command.