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

French title: Algorithmes efficaces en calcul formel.

In charge: F. Chyzak

Important note: dates of sessions from December 9, 2021 on have been changed.

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.

In a second part of the course, we enter more specialized, active research domains, including symbolic summation and integration. The symbolic summation and integration methods that we describe have become unexpendable computation tools for physics, the numeric evaluation of special functions, the modelling of combinatorial problems, and more broadly for difficult questions of formulas simplification.

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 (2-13-1), Codes correcteurs d'erreurs et applications à la cryptographie (2-13-2), Techniques en cryptographie et cryptanalyse (2-12-1), Algorithmes arithmétiques pour la cryptologie (2-12-2) and Analyse d'algorithmes (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 Thursdays, 16:15-19:15, room 1002.

Alin Bostan (15h), Frédéric Chyzak (15h), Pierre Lairez (9h), Vincent Neiger (9h).

Summary: French spoken, unless English requested; all slides in English; book in French will not be translated.

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. 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. (The year 2020-2021 was fully taught in English.)

Except for students who will officially chose 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. Documents allowed during examinations will be the book and personal notes by the student.

The course is breakable, so that student may elect to not attend the second period (thus validating only half of the ECTS), but we strongly discommend attending the second period only.

Book: AECF

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

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

Fast evaluation and interpolation. Gcd and resultant. (Chapters 5 and 6) Slides Exercises

Fast computations with series. D-finite power series. (Chapters 3 and 14) Slides Exercises

Linear recurrences: constant coefficients and rational functions, polynomial coefficients: n-th term, first n terms. (Chapters 4 and 15) Slides Exercices

Computations with polynomial matrices: introduction and motivations. Slides Exercices Solutions to last week's exercises

Polynomial factorization over finite fields. (Chap 19) Slides Exercises Solutions to last week's exercises

Exercises session.

Some rules: The consultation of static data (lessons and personal notes) 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.

First-period exam of 2013 (in French)

Polynomial matrices: fast approximation and interpolation, quasi-linear GCD. Slides Exercises

Using approximation and interpolation for kernel bases, quasi-linear gcd, and other applications. Slides Exercises

Factorization over the integers, lattice reduction. (Chapters 20 and 21) Slides Notebook sagemath

Polynomial and rational solutions of linear ODEs and OREs. (Chapters 16 and 17) Slides Exercises

Gosper's, Zeilberger's, Almkvist and Zeilberger's algorithms. Algorithms by compact forms. (Chapter 29 and part of Chapter 16) Slides Exercise

Binomial sums and diagonals. Slides Exercices Maple worksheet

Computer algebra for combinatorics. Slides

Exercises session.

General works:

• D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, Springer Verlag, 2nd edition, 1996.
• 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.

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/.