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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. ObjectivesComputer 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 computeralgebra algorithms to work with polynomials, series, and matrices, so as to achieve in many cases quasioptimal complexity bounds. Such algorithms are largely used in practice in computeralgebra systems, but as well in several other modern algorithmic domains that rely on algebraic techniques, such as cryptography, multivariate cryptoanalysis, and errorcorrecting 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 (2131), Codes correcteurs d'erreurs et applications à la cryptographie (2132), Techniques en cryptographie et cryptanalyse (2121), Algorithmes arithmétiques pour la cryptologie (2122) and Analyse d'algorithmes (215). Organization for 20212022This 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. Time and location for 20212022On Thursdays, 16:1519:15, room 1002. ProfessorsLanguageSummary: 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 nonFrenchspeaking 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 20202021 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 Lecture 1. 23/09F. ChyzakGeneral presentation of the course. Fast polynomial multiplication. (Chapters 1 and 2) Slides Exercises Lecture 2. 30/09A. BostanLecture 3. 07/10A. BostanLecture 4. 14/10F. ChyzakLecture 5. 21/10F. ChyzakLinear recurrences: constant coefficients and rational functions, polynomial coefficients: nth term, first n terms. (Chapters 4 and 15) Slides Exercices Lecture 6. 28/10V. NeigerComputations with polynomial matrices: introduction and motivations. Slides Exercices Solutions to last week's exercises Lecture 7. 04/11P. LairezPolynomial factorization over finite fields. (Chap 19) Slides Exercises Solutions to last week's exercises ATTENTION: NO LECTURE ON 11/11!Lecture 8. 18/11A. BostanATTENTION: NO LECTURE ON 25/11!FIRSTPERIOD EXAM ON 2/12Some 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. ATTENTION: NO LECTURE ON 09/12!Lecture 9. 16/12V. NeigerLecture 10. 06/01V. NeigerUsing approximation and interpolation for kernel bases, quasilinear gcd, and other applications. Slides Exercises Lecture 11. 13/01P. LairezFactorization over the integers, lattice reduction. (Chapters 20 and 21) Slides Notebook sagemath Lecture 12. 20/01F. ChyzakLecture 13. 27/01F. ChyzakGosper's, Zeilberger's, Almkvist and Zeilberger's algorithms. Algorithms by compact forms. (Chapter 29 and part of Chapter 16) Slides Exercise ATTENTION: NO LECTURE ON 3/02!Lecture 14. 10/02P. LairezBinomial sums and diagonals. Slides Exercices Maple worksheet Lecture 15. 17/02A. BostanComputer algebra for combinatorics. Slides Lecture 16. 24/02A. BostanExercises session. ATTENTION: NO LECTURE ON 03/03!SECONDPERIOD EXAM ON 10/03BibliographyGeneral works:
The book emanating from our lectures over the past years:
Pedagogical team
