Efficient Algorithms in Computer Algebra (48h, 6 ECTS)

French title: Algorithmes efficaces en calcul formel.

Professors for 2024-2025: Alin Bostan, Pierre Lairez, Marc Mezzarobba, Vincent Neiger.

Computer Algebra (also known as Symbolic Computation, or Calcul formel in French) consists in developing computer representations and manipulations of mathematical objects in an exact way, in contrast with traditional scientific computing, for example. As a counterpart of such exact algebraic representations, computation times are often large and memory requirements are often huge if one employs naive algorithms. In this course, we introduce basic computer-algebra algorithms to work with polynomials, series, and matrices, so as to achieve in many cases quasi-optimal complexity bounds. Such algorithms are largely used in practice in computer-algebra systems, but as well in several other modern algorithmic domains that rely on algebraic techniques, such as cryptography, multivariate cryptoanalysis, and error-correcting codes.

This course is particularly well suited to those who wish to rigorously understand the algorithmic foundations of algebraic calculation and their general principles, as a supplement of the courses Systèmes polynomiaux, calcul formel et applications (Polynomial Systems, computer algebra, and applications 2-13-1), Codes correcteurs d'erreurs et applications à la cryptographie (Error correcting codes and applications to cryptography 2-13-2), Techniques en cryptographie et cryptanalyse (Techniques in cryptography and cryptanalysis 2-12-1), Algorithmes arithmétiques pour la cryptologie (Arithmetic algorithms for cryptology 2-12-2) and Analyse d'algorithmes (Analysis of algorithms 2-15).

This webpage holds the contents of the whole course and will be updated on a weekly basis to integrate various notes and documents, and potentially to reflect evolutions based on what could be presented during lectures.

On Mondays, 16:15-19:15, room 1004. Exception: course on Thursday 14 November, 16:15-19:15, room 1004.

Summary: English spoken if requested, otherwise French spoken; all slides in English; book in French.

In line with the MPRI terminology, the course is a module “English upon request”, meaning that (oral) lectures will be in French, unless at least one non-French-speaking student requests English (in most recent years, the course has been fully taught in English). Nevertheless, our main reference book was written in French and will not be translated. In all lectures, whatever the spoken language, slides will be written in English. In all cases, students will be allowed to take their exams in French or English.

Except for students who will officially choose to quit the course after the first half (“breakable module”), the global mark will be an average between the marks for the first and second halves of the course.

The course is breakable, so that students may elect not to attend the second period (thus validating only half of the ECTS). We strongly recommend against attending the second period only, without the first one.

Book: AECF

Other useful references can be found at the end of this webpage.

General presentation of the course. Fast polynomial multiplication. (Chapters 1 and 2)

Slides

Dense linear algebra: from Gauss to Strassen. (Chapter 8)

Slides SageMath illustrations Exercises

D-finite power series. (Chapter 14)

Slides (no-overlay version) Exercises

Fast evaluation and interpolation. (Chapter 5)

Slides Exercices

Gcd, resultant. Fast computations with power series via Newton iteration. (Chapters 3 and 6)

Slides Exercises

Linear recurrences with constant coefficients and rational functions: n-th coefficient + fast composition of power series

Slides Slides (no-overlay version) Exercises

Computations with polynomial matrices: introduction and motivations.

Slides Exercises 3 and 6 Solutions

Tutorial/Exercises session.

Exercises

Some rules: The consultation of static data (lecture slides, personal notes, computer algebra books, ...) on an electronic device is authorized, provided that these devices are not connected to any network. Consultation of the course and personal notes on paper is authorized.

Fast approximant bases and applications.

Slides

Exercises: many exercises are presented all along these slides. For some, a solution is presented in the slides; some others can be done via SageMath. If you have any question or issue with these exercises, please contact the lecturer by email.

Polynomial factorization over finite fields. (Chap 19)

Slides

Polynomial factorization over the integers (Chapters 20 and 21)

Slides

Lattice reduction. Experimental mathematics.

Slides

Binomial sums.

Slides

Exercise

Linear recurrences with polynomial coefficients: n-th term, first n terms. (Chapters 4 and 15)

Slides (no-overlays, annotatable), Exercises

Solutions of linear differential equations. (Chapter 17)

Slides Slides (no overlays)

Computer algebra for combinatorics.

Slides

In recent years, the second-period exam consisted in a presentation of a research article, followed by some questions by the course teachers. The research article was chosen by the student a few weeks before, from a list provided by the teachers.

General works:

• J. von zur Gathen, J. Gerhard, Modern Computer Algebra, Cambridge University Press, 1999.
• M. Petkovsek, W. Wilf, D. Zeilberger, A=B, A. K. Peters, 1996.
• K. O. Geddes, S. R. Czapor, G. Labahn, Algorithms for Computer Algebra, Kluwer Academic Publishers, 1992.
• D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, Springer Verlag, 2nd edition, 1996.

The book emanating from our lectures over the past years:

• A. Bostan, F. Chyzak, M. Giusti, R. Lebreton, G. Lecerf, B. Salvy and É. Schost, Algorithmes Efficaces en Calcul Formel. Printed by CreateSpace. Palaiseau: F. Chyzak (self-ed.), 2017. Also available for free in electronic format. https://hal.archives-ouvertes.fr/AECF/.