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 19.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.

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Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as W = ⊕g∈GWg with WaW b ⊂ W ab , and where each Wg is a one dimensional K-vector space. It is also assumed that W has no monomial which is a zero divisor which means that for each pair of nonzero elements wa ∈ Wa, w b ∈ W b , wa • w b = 0. We also demand that W has a multiplicative identity element. We focus in the case where G is a finite abelian group and K = C or K = R. In this article, using methods of group cohomology, we classify all associative G-graded twisted algebras in the case G is a finite abelian group. On the other hand, by generalizing some of the arguments developed in [1] we present a classification of all G-graded twisted algebras that satisfy certain symmetry condition.