Symbolic Computation Group

Many fundamental concepts in mathematics are defined in terms of limits and it is desirable for computer algebra systems to be able to compute them. However, limits of functions, limits of secants or topological closures are, by essence, hard to compute in an algorithmic fashion, say by doing finitely many rational operations on polynomials or matrices over the usual coefficient fields of symbolic computation. This is why a computer algebra system like Maple is not capable of computing limits of rational functions in more than two variables while it can perform highly sophisticated algebraic computations like solving (formally) a system of partial differential equations.

In our ISSAC 2016, we proposed an algorithm for computing limits of real multivariate rational functions at an isolated zero of the denominator. Today's talk extends this work. First, we present an algorithm for determining the real branches of a space curve about one of its point. This is a core routine for computing limits of real multivariate rational functions as well as for addressing topological questions like whether a point belongs to the closure of a CAD cell. To this end, we revisit the Hensel-Sasaki construction and illustrate its usage with the PowerSeries library. In a second part, we discuss what remains to be done to obtain a general algorithm for computing limits of real multivariate rational functions.

This is a joint work with Parisa Alvandi, Masoud Ataei and Marc Moreno Maza.