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Computing Limit Points of Quasi-Components of Regular Chains and Its Applications

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Computing Limit Points of Quasi-Components of Regular Chains and Its Applications Western University Scholarship@Western Electronic Thesis and Dissertation Repository 5-11-2017 12:00 AM Computing Limit Points of Quasi-components of Regular Chains and its Applications Parisa Alvandi The University of Western Ontario Supervisor Dr. Marc Moreno Maza The University of Western Ontario Graduate Program in Computer Science A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Parisa Alvandi 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Theory and Algorithms Commons Recommended Citation Alvandi, Parisa, "Computing Limit Points of Quasi-components of Regular Chains and its Applications" (2017). Electronic Thesis and Dissertation Repository. 4565. https://ir.lib.uwo.ca/etd/4565 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract Computing limits is a fundamental task in mathematics and different mathematical con- cepts 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 so- lutions 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 di- mension 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 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. Keywords: Regular chains, quasi-components, limit points, tangent cone, limit of multi- variate rational functions, extended Hensel construction. i Acknowlegements I would like to thank all the people who contributed in some way to the work described in this thesis. First, I would like to express my sincere appreciation and gratitude to professor Marc Moreno Maza for his guidance during my research. His support and inspiring suggestions have been precious for the accomplishments of this thesis content. During my study at The University of Western Ontario, he contributed to a rewarding graduate school experience by supporting my attendance at various conferences, engaging me in new ideas, and demanding a high quality of work in all my endeavors. Additionally, I would like to thank my committee members Professor Agnes Szanto, Pro- fessor Jan Minac, Professor Lila Kari, and Professor Olga Veksler for their interest in my work. Every result described in this thesis was accomplished with the help and support of fellow labmates and collaborators. I feel honoured to collaborate with my brilliant, insightful co- authors: Professor Amir Hashemi, Professor Eric´ Schost, Dr. Changbo Chen, Dr. Paul Vrbik, Dr. Masoud Ataei, and Mahsa Kazemi. Finally, I would like to acknowledge friends and family who supported me during my time here. First and foremost, I would like to thank my mother and father , Fariba and Hossein, and my brothers, Mostafa, Mohammad, and Reza, for their great love, support and sacrifices. I also would like to thank my friends Sugi Magesan and Elham Karami for supporting me all along this way. I am also grateful for their help in proofreading some chapters of my thesis. ii Contents Abstract i Acknowlegements i List of Algorithms vi List of Figures vii List of Tables viii 1 Overview 1 1.1 Goals . 8 1.2 Thesis accomplishments . 9 1.2.1 Computing limit points of quasi-components of regular chains of di- mension one. 10 1.2.2 Improving the extended Hensel construction. 10 1.2.3 Computing the real limit points of the quasi-component of a regular chain of dimension one. 11 1.2.4 Studying regular chains under changes of coordinates. 11 1.2.5 Introducing new tools for computing tangent cones of space curves. 11 1.2.6 Computing limit of real multivariate rational functions. 11 1.2.7 Separating the real and complex branches of space curves. 12 1.2.8 Thesis contribution in RegularChains and PowerSeries libraries. 13 1.3 Contribution statement . 14 1.4 Thesis outline . 14 2 Background and Related Work 16 2.1 Solving polynomial systems . 16 2.1.1 Limit points . 21 2.2 Power series and Puiseux expansions . 21 2.3 The problem and related work . 23 3 Extended Hensel Construction 26 3.1 Introduction . 26 3.2 Extended Hensel construction . 28 3.2.1 Extended Hensel construction of multivariate polynomials . 32 3.2.2 Complete factorization in C(hY∗i)[X] .................. 34 iii 3.3 On the Yun-Moses polynomials . 34 3.3.1 Computing the Wλ ............................ 37 3.3.2 Complexity analysis . 38 3.4 Lifting the factors . 39 3.4.1 Complexity analysis . 41 3.5 Experimentation . 41 4 Computing Limit Points via Puiseux Series Expansions 44 4.1 Introduction . 44 4.2 Preliminaries . 47 4.2.1 Basic techniques . 49 4.3 Puiseux expansions of a regular chain . 52 4.4 Puiseux parametrization in finite accuracy . 54 4.5 Computing in finite accuracy . 56 4.6 Accuracy estimates . 60 4.7 Algorithm . 62 4.8 Experimentation . 64 4.9 Concluding remarks . 65 5 Real Limit Points of Space Curves 66 5.1 Introduction . 66 5.2 Real limit points . 68 5.2.1 Real branches of bivariate polynomials . 70 5.2.2 Real branches of space curves . 73 5.3 Experimentation . 74 6 Computing Limit Points via Changes of Coordinates 76 6.1 Introduction . 76 6.2 Preliminaries . 77 6.3 Algorithm for linear change of coordinates . 78 6.3.1 The PALGIE algorithm for the prime case . 80 6.3.2 Regularity test in IsRegular(p; C; R)................... 89 6.3.3 The PALGIE algorithm for linear change of coordinates . 89 6.4 Noether normalization and regular chains . 90 6.5 Applications of random linear changes of coordinates . 91 6.6 On the computation of lim(W(T)) and sat(T)................... 93 6.7 Conclusion . 99 7 Tangent Cones of Space Curves 100 7.1 Introduction . 100 7.2 Preliminaries . 101 7.2.1 Tangent cone of a space curve . 102 7.2.2 Regular chains . 103 7.3 Computing intersection multiplicities in higher dimension . 104 7.4 Computing tangent lines as limits of secants . 105 iv 7.4.1 An algorithmic principle . 105 7.4.2 Algorithm . 107 7.4.3 Equations of tangent cones . 109 7.4.4 Examples . 110 7.5 Conclusion . 113 8 Computing Limits of Multivariate Rational Functions 115 8.1 Introduction . 115 8.2 Preliminaries . 118 8.2.1 Lagrange multipliers . 118 8.2.2 Regular chain theory . 119 8.2.3 Parametric polynomial systems . 120 8.2.4 Triangular decomposition of semi-algebraic sets . 121 8.2.5 Puiseux series . 123 8.3 Basic lemmas . 123 8.4 Main Algorithm . 126 8.5 Optimizations . 130 8.6 Limits of multivariate rational functions: general case . 131 8.7 Experimentation . 134 8.8 Conclusion . 135 9 Conclusion 137 9.1 Computing limit points of quasi-components of regular chains . 137 9.2 Computing Puiseux expansions of bivariate polynomials . 138 9.3 Computing tangent cones of space curves at their singular points . 138 9.4 Computing limits of real multivariate rational functions . 139 v List of Algorithms 1 EHC Lift ...................................... 32 2 NonzeroTerm ................................... 56 3 NewtonPuiseux .................................. 57 4 LimitPointsAtZero ................................. 63 5 LimitPoints ..................................... 63 6 RealPuiseuxExpansions ............................. 72 7 RealRegularChainBranches ........................... 74 8 IsRegular(p; C; R)................................. 81 9 Saturate(C; H; R).................................. 82 10 Extend(C; D; R).................................. 82 11 EnsureRank(p; R; C; R).............................. 83 12 EnsureLeadingCoefficient(p; v; R; C; R)..................... 83 − 13 Gcdn(q; p; v; Cv ; R; C; R).............................. 84 14 Palgie(C; R; R)................................... 85 15 Closure(T)..................................... 96 16 TangentCone ...................................
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