On the Extended Hensel Construction and its application to the computation of real limit points

Abstract The Extended Hensel Construction (EHC) is a procedure which, for an input bivariate polynomial with complex coefficients, can serve the same purpose as the Newton-Puiseux algorithm, and, for the multivariate case, can be seen as an effective variant of Jung-Abhyankar Theorem. We show that the EHC requires only linear algebra and univariate polynomial arithmetic. We deduce complexity estimates and report on a software implementation together with experimental results. This work is motivated and illustrated by two applications. The first one is the computation of real branches of space curves. The second one is the computation of limits of real multivariate rational function. For the latter, we present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point p which is an isolated zero of the denominator of q. When the limit exists, the algorithm computes it, without making any assumption on the number of variables. A process, which extends the work of Cadavid, Molina and Velez, reduces the multivariate setting to computing limits of bivariate rational functions. By using regular chain theory and triangular decomposition of semi-algebraic systems, we avoid the computation of singular loci and the decomposition of algebraic sets into irreducible components.