Doctor of Philosophy
Dr. Marc Moreno Maza
Computing limit is a fundamental task in mathematics and different mathematical concepts are defined in terms of limit computations. Among these mathematical concepts, we are interested in three different types of limit computations: first, computing the limit points of solutions of polynomial systems represented by regular chains, second, computing tangent cones of space curves at their singular points which can be viewed as computing limit of secant lines, and third, computing the limit of real multivariate rational functions.
For computing the limit of solutions of polynomial systems represented by regular chains, we present two different methods based on Puiseux series expansions and linear changes of coordinates. The first method, which is based on Puiseux series expansions, addresses the problem of computing real and complex limit points corresponding to regular chains of dimension one. The second method studies regular chains under changes of coordinates. It especially computes the limit points corresponding to regular chains of dimension higher than one for some cases. we consider strategies where these changes of coordinates can be either generic or guided by the input.
For computing the Puiseux parametrizations corresponding to regular chains of dimension one, we rely on extended Hensel construction (EHC). The Extended Hensel Construction 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.
We also outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a Standard basis.
We also present an algorithm for determining the existence of the limit of a real multivariate rational function q at a given point 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 V´elez, 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.
Alvandi, Parisa, "Computing Limit Points of Quasi-components of Regular Chains and its Applications" (2017). Electronic Thesis and Dissertation Repository. 4565.