Real valuations and the limits of multivariate rational functions

摘要

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 is an element of R [x(1) ...., x(n)] and (a(1) ...,a(n)) is an element of R-n, the (finite) limit of the rational function f/g at (a(1), ..., a(n)) does not exist if and only if (1) there exists a sequence u(1)(x),..., un(x) of polynomials over R in one variable x such that u(i)(0) = a(i) for i = 1, ...,n, g(u(1)(x)) not equal 0, but lim(x -> 0) f(u(1)(x) ..., u(n) (x))/g(u(1)(x), ...,u(n)(x)) = infinity; or (2) there exist two sequences