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

French title: Algorithmes efficaces en calcul formel.

Professors for 2023-2024: 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 Thursdays, 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)

Documents

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

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

Fast computations with power series. Newton iteration. (Chapters 3 and 6)

D-finite power series. (Chapter 14)

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

Solutions of linear differential equations. (Chapter 17)

Tutorial/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.

Computations with polynomial matrices: introduction and motivations.

Polynomial matrices: fast Hermite-Padé approximation and vector interpolation.

A selection of applications of polynomial matrix computations

Polynomial factorization over finite fields. (Chap 19)

Factorization over the integers. (Chapters 20 and 21)

Lattice reduction. Application to factorisation and guessing.

Binomial sums.

Computer algebra for combinatorics.

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/.