**Context: **
This is a research course
from the Master 2 in Computer Science program, of the
Computer Science Department
of the
Ecole Normale Supérieure in Lyon.

The lectures will be delivered
in the academic year 2022-2023 (for 15 x 2h and 5 ECTS).

**Instructors: **
Alin Bostan,
Bruno Salvy,
Gilles Villard.

**Description: **
Computer Algebra, or Symbolic Computation, consists of representing and
manipulating mathematical objects on a computer in an exact way, in contrast
with the traditional scientific computing. In this course, we present a very
active research topic in Computer Algebra – the development of
*algorithms for symbolic summation and symbolic integration*. The
emergence of these algorithms has put an important family of computational
tools in the hands of discrete mathematicians, and many results that were
inaccessible in other ways have been found and proved by computer methods.
These algorithms are also nowadays unavoidable in physics, for the numerical
evaluation of special functions, and more broadly for the difficult questions
of simplification of formulas.

**Prerequisites: **
Preferably the audience should have followed a level M1 introductory course in Computer Algebra.
In particular, notions on **basic algebraic
objects** (polynomials, matrices, power series) and **basic algebraic
algorithms** (Gaussian elimination, Euclidean algorithm) will be helpful.

**Objective: **
We will explain, among other things, how modern computer algebra answers the
following open problem from the book *The Art of Computer Programming,
Volume 1: Fundamental Algorithms* by Donald E. Knuth, Addison Wesley,
Reading, Massachusetts, 1968:

**[50] Develop computer programs for simplifying sums that
involve binomial coefficients.**
As an example of a recent combinatorial application of the algorithms
presented in this course, we will describe a computer-driven proof of the
following fact: the counting generating function of walks with prescribed
length, with allowed steps in the set X, starting from
the origin and confined to the quarter plane is:

• *algebraic* if X = {E, W, NE, SW} (*Gessel walks*, see figure below);

• *transcendental* if X = {N, S, NE, NW} (*trident walks*).

**Detailed plan of the lectures: **

- 0- Introduction (Bruno Salvy).
- 1- Algorithms for polynomial matrices.
- Matrix and polynomial computations (Gilles Villard).
- Matrix and polynomial computations / Padé-Hermite approximation (Gilles Villard).
- Padé-Hermite approximation / Alternative to Gaussian elimination (Gilles Villard).
- Alternative to Gaussian elimination / bivariate resultants (Gilles Villard).

- 2- Algorithms for recurrence and differential equations.
- P-recursive sequences and D-finite functions: closure properties (Alin Bostan).
- Algorithmic guessing of algebraic and differential equations (Alin Bostan).
- Polynomial, rational and hypergeometric solutions of linear differential/recurrence equations (Alin Bostan).

- 3- Algorithms for symbolic summation and integration.
- Creative Telescoping — from 1G to 4G algorithms (Alin Bostan).
- Definite integration of multivariate rational functions (Bruno Salvy).
- Multiple binomial sums (Bruno Salvy).
- Combinatorial applications (Alin Bostan).
- Definite integration of general D-finite functions (Bruno Salvy).

**Validation: ** There will
be a mid-term exam or a homework assignment
and a written final exam.

**Bibliography:
**

**ARTICLES (click on the title to get the pdf): **

•
G. Villard: *On computing the resultant of generic bivariate polynomials.* ISSAC'18, 391‐398, ACM Press, 2018.

•
W. Zhou, G. Labahn: *Efficient algorithms for order basis computation.* J. Symbolic Comput. 47 (2012), no. 7, 793‐819.

• A. Storjohann: *High-order lifting and integrality certification.* J. Symbolic Comput. 36 (2003), no. 3-4, 613‐648.

• W. Zhou, G. Labahn, A. Storjohann: *A deterministic algorithm for inverting a polynomial matrix.* J. Complexity 31 (2015), no. 2, 162‐173.

• G. Labahn, V. Neiger, W. Zhou: *Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix.* J. Complexity 42 (2017), 44‐71.

• A. Benoit, A. Bostan, J. van der Hoeven:
*Quasi-optimal multiplication of linear differential operators.*
FOCS 2012, 524‐530, IEEE Computer Soc., 2012.

• A. Bostan, F. Chyzak, Z. Li, B. Salvy:
*Fast computation of common left multiples of linear ordinary differential operators.* ISSAC'12, 99‐106, ACM Press, 2012.

• J. van der Hoeven:
*On the complexity of skew arithmetic.*
Appl. Algebra Engrg. Comm. Comput. 27 (2016), no. 2, 105‐122.

•
A. Bostan, T. Cluzeau, B. Salvy:
*Fast algorithms for polynomial solutions of linear differential equations.* ISSAC'05, 45‐52, ACM Press, 2005.

•
A. Bostan, F. Chyzak, T. Cluzeau, B. Salvy:
*Low complexity algorithms for linear recurrences.*
ISSAC'06, 31‐38, ACM Press, 2006.

•
F. Chyzak:
*An extension of Zeilberger's fast algorithm to general holonomic functions.*
Discrete Math. 217 (2000), no. 1-3, 115‐134.

• A. Bostan,
S. Chen, F. Chyzak, Z. Li:
*Complexity of Creative Telescoping for Bivariate Rational Functions.*
ISSAC'10, pp. 203‐210, ACM Press, 2010.

• A. Bostan, P. Lairez, B. Salvy:
*Creative telescoping for rational functions using the Griffiths-Dwork method.*
ISSAC'13, pp. 93‐100, ACM Press, 2013.

• A. Bostan, P. Lairez, B. Salvy:
*Multiple binomial sums.*
J. Symbolic Comput. 80 (2017), part 2, 351‐386.

•
A. Bostan, F. Chyzak, P. Lairez, B. Salvy:
*Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions.*
ISSAC'18, pp. 95‐102, ACM Press, 2018.

**BOOKS (click on the title to go to the book's website and on the image to get the pdf): **

• M. Petkovsek, W. Wilf, D. Zeilberger, *A=B*, A. K. Peters, 1996.

• A. Bostan, F. Chyzak, M. Giusti, R. Lebreton, G. Lecerf, B. Salvy, É. Schost,
*Algorithmes Efficaces en Calcul Formel*, CreateSpace, 2017.

• J. von zur Gathen, J. Gerhard, *
Modern Computer Algebra*, Cambridge
University Press, 1999.