Lecture Materials

	Given	Topic	Slides
Lecture 1	Tuesday May 13	Motivating examples	slides
Lecture 2	Thursday May 15	From convergent to formal power series	slides
Lecture 3	Tuesday May 20	Weierstrass Preparation Theorem (1/2)	slides
Lecture 4	Thursday May 22	Weierstrass Preparation Theorem (2/2)	slides
Lecture 5	Tuesday May 27	Weierstrass Preparation Theorem: UPoPs case	slides
Lecture 6	Wednesday May 28	Properties of Power Series Rings	slides
Lecture 7	Tuesday June 3	Laurent and Puiseux series
Lecture 8	Thursday June 5	Puiseux theorem and consequences
Lecture 9	Tuesday June 10	Regular chain theory (3 hours)
Lecture 10	Thursday June 12	Limit Points of constructible sets
Lecture 11	Tuesday June 17	Limit of multivariate rational functions
Lecture 12	Wednesday June 23	Tangent Cones
Lecture 13	Tuesday June 24	Project presentations
Lecture 14	Thursday June 26	Project presentations

Some research papers used in the lectures are listed below

• R. P. Brent and H. T. Kung: Fast Algorithms for Manipulating Formal Power Series
• TZEE-CHAR KUO: Generalized Newton-Puiseux Theory and Hensel's Lemma in C[[x, y]]
• Michael Greenblatt: Adapted Coordinates in Two Dimensions and a Proof of Puiseux’s Theorem
• H. T. Kung and J. F. Traub: All algebraic functions can be computed fast
• P.G. Walsh: ON THE COMPLEXITY OF RATIONAL PUISEUX EXPANSION
• Adrien Poteaux · Marc Rybowicz: Complexity Bounds for the rational Newton-Puiseux Algorithm over Finite Fields
• Dominique Duval: Rational Puiseux expansions
• Joris van der Hoeven: A new zero-test for formal power series
• Joris van der Hoeven: Relax, but Don’t be Too Lazy
• J. Rafael Sendra, David Sevilla, and Carlos Villarino: Some Results on the Surjectivity of Surface Parametrizations
• Gary R. Lawlor: A L'hospital's rule for multivariable functions
• V. V. Ivlev, I. A. Shilin: On a generalization of l'Hopital's rule for multivariable functions
• JESSE ELLIOTT: Factoring formal power series over principal ideal domains

Some research slides used in the lectures are listed below

• Tutorial on the theory and practice of regular chains

Some Maple programs used in the lectures are listed below

• The MultivariatePowerSeries library web site
• The MultivariatePowerSeries demo
• The MultivariatePowerSeries Installation Guide

Some reference books are listed below

• H. JEROME KEISLER : FOUNDATIONS OF INFINITESIMAL CALCULUS
• P_Dolbeault: Analyse Complexe
• James D. Lewis: Complex variables
• Modern Computer Algebra
• Ideals, Varieties, and Algorithms
• Using Algebraic Geometry
• Algorithms in Real Algebraic Geometry

Computer algebra systems and related software used in this course

• Maple
• AXIOM
• Julia
• CilkPlus
• The RegularChains Library in Maple
• Basic Polynomial Algebra Subprograms (BPAS)

Quizzes and Projects

Tentative Quiz and Project Schedule

	Given	Due	Weight
Quiz 1	Tuesday June 3	9:30-10:00	15%
Quiz 2	Tuesday June 17	9:30-10:00	15%
Project selection	Thursday June 5	3:00 - 6:00pm
Project presentations	June 24 and 26	9:30 - 11:00	70%

Teaching Crew and Hours

• Instructor
  Marc Moreno Maza
  email: moreno@csd.uwo.ca
  
  Tuesday lecture: 9:30-11:00
  Zoom meeting Passcode required
  Thursday lecture: 9:30-11:00
  Zoom meeting Passcode required
  

CS9646, Summer 2025
Algorithms for Multivariate Power Series and their Application to Symbolic analysis
Course Outline

Course Description

Many fundamental concepts in mathematics are defined in terms of limits and it is highly desirable for computer algebra to implement those concepts. However, limits are, by essence, hard to compute, or even not computable in an algorithmic fashion, say by doing finitely many rational operations on polynomials or matrices.

In this course, we shall see how various types of limits can be computed by means of algebraic calculations. Examples will cover the Zariski closure of a constructible set, the tangent cone of an algebraic set at one of its singular points, and the limit of a real multivariate rational function at one of its poles.

The first half of this course will consist of a presentation of the underlying mathematical concepts (multivariate power series and Puiseux series) and algorithms (factorization of polynomials over multivariate power series rings, including the Extended Hensel Construction) as well as software demonstration powered by the RegularChains and MultivariatePowerSeries libraries in Maple.

Instructor

Marc Moreno Maza
Email: moreno@csd.uwo.ca
Office: Middlesex College 327

Course Materials

There is no specific textbook for this course. Lecture notes, suggested readings and supplementary materials are available electronically on the course website.

Course Website

Students should check the course website http://www.csd.uwo.ca/courses/CS9646 on a regular basis for news and updates. These are the primary method by which information will be disseminated to all students in the class. The missing of critical information due to your failure to check the course website cannot be used as a basis for appeal.

Class Schedule

Tuesdays and Thursdays from 9:30 to 11:00 from May 13 to June 26.

Course Work Evaluation

The overall course grade, out of 100, will be calculated as follows:

• 2 in-class quizzes about key concepts and principles, each worth 15%, the sum of which will constitute 30% of the final grade;
• 1 course project worth 70%;

A CS9646 project topic is chosen by the student from a list of topics proposed by the instructor. Project topics will be posted by May 29 and each student must choose a project topic by June 5. The projects will be presented in class by the students during the last two weeks of classes. Each presentation will consist of a 20 minute talk followed by questions for 5 minutes.

There is no midterm examination and no final examination.

It is Faculty of Science policy that a student who chooses to write a test or exam deems themselves fit enough to do so, and the student must accept the mark obtained. Claims of medical, physical, or emotional distress after the fact will not be considered.

Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at this website:

http://www.uwo.ca/univsec/pdf/academic_policies/appeals/scholastic_discipline_undergrad.pdf.

Computer-marked, multiple-choice tests and exams may be subject to submission for similarity review by software that will check for unusual coincidences in answer patterns that may indicate cheating.

Every effort will be made to have quizzes marked and handed back within 2 weeks of the hand-in date.

Tentative Quiz and Project Schedule

	Given	Due	Weight
Quiz 1	Friday. May. 21	10:30-11:00	15%
Quiz 2	Friday June 16	10:30-11:00	15%
Project selection	Wednesday June 2	3:00 - 6:00pm
Project presentations	June 23 and 25	10:30 - 12:00	70%

• If for any reason the schedule cannot be adhered to, the marks will be pro-rated.
• No electronic devices may be in your possession during tests.
• All quizzes are close-book. A reference sheet (provided by the instructor) is allowed for the quizzes.
• In case of questions regarding the marks, please note that no quizzes will be accepted for re-marking later than two weeks after they have been marked. We reserve the right not to re-mark quizzes or exams that have been written in pencil.
• All students are expected to attend all classes. A student found to be missing a large number of classes without acceptable reasons risks being denied a passing grade.

Missed Test or Final Exam

If you are unable to meet a course requirement due to illness or other serious circumstances, you must provide valid medical or supporting documentation to the Academic Counselling Office of your home faculty as soon as possible. The University Policy on Accommodation Consideration for Student Absences can be found here.

If you are a science student, the Academic Counselling Office of the Faculty of Science can be contacted at 519-661-3040 or scibmsac@uwo.ca. See their Their web site.

A student requiring academic accommodation due to illness must use the Student Medical certificate when visiting an off-campus medical facility.

For further information, please consult the university's medical illness policy. and the University' policy on academic accommodation for student with disabilities.

Ethical Conduct

Scholastic offences are taken seriously and students are directed to read the appropriate policy, specifically, the definition of what constitutes a Scholastic Offence, at the following web site.

All assignments are individual assignments. You may discuss approaches to problems among yourselves. However, the actual details of the work (assignment coding, answers to concept questions, etc.) must be your individual effort. Assignments that are judged to be the result of academic dishonesty will, for the student's first offence. You are responsible for reading and respecting the Computer Science Department's policy on Rules of Ethical Conduct and Scholastic Offenses.

All required papers may be subject to submission for textual similarity review to the commercial plagiarism detection software under license to the University for the detection of plagiarism. All papers submitted for such checking will be included as source documents in the reference database for the purpose of detecting plagiarism of papers subsequently submitted to the system. Use of the service is subject to the licensing agreement, currently between The University of Western Ontario and Turnitin.com.

Computer-marked multiple-choice tests and/or exams may be subject to submission for similarity review by software that will check for unusual coincidences in answer patterns that may indicate cheating.

Accessibility

Please contact the course instructor if you require lecture or printed material in an alternate format or if any other arrangements can make this course more accessible to you. You may also wish to contact Services for Students with Disabilities (SSD) at 661-2111 ext. 82147 if you have questions regarding accommodation.

Mental Health

All students who are in emotional/mental distress should refer to Mental Health@Western. for a complete list of options about how to obtain help.

Support Services

Learning-skills counsellors at the Student Development Centre. are ready to help you improve your learning skills. They offer presentations on strategies for improving time management, multiple-choice exam preparation/writing, textbook reading, and more. Individual support is offered throughout the Fall/Winter terms in the drop-in Learning Help Centre, and year-round through individual counselling. See also the services provided by the University Students’ Council.

The website for Registrarial Services.

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