## Section: New Results

### Computer Algebra

Participants : Alexandre Benoit, Alin Bostan, Frédéric Chyzak, Philippe Flajolet, Manuel Kauers, Nicolas Le Roux, Marc Mezzarobba, Bruno Salvy.

One of our main themes of research is the use of linear differential equations as a data structure for the automatic manipulation of special functions. This concerns various operations, including definite integration. In this vein, J. Borwein, and B. Salvy give a proof [4] of a conjecture on moments of powers of the Bessel function that had attracted the attention of mathematical physicists. Not only is the proof effective, but it gives an efficient derivation of formulas for the computations of these moments.

Newton iteration can be put to use in the context of power series of nonlinear differential equations. A. Bostan, F. Morain (project-team Tanc), B. Salvy, and É. Schost apply this idea in [6] to a problem in elliptic curve cryptography. They introduce an algorithm that computes a special rational morphism between two elliptic curves; its complexity is quasi-linear with respect to the degree of the morphism. The new algorithm allows to speed-up a crucial step of the Schoof-Elkies-Atkin algorithm for determining the cardinality of an elliptic curve over a finite field.

Composition of power series is a fundamental operation for which no quasi-optimal algorithm is known except for special cases. The class of special cases has been vastly extended by A. Bostan, B. Salvy, and É. Schost in [16] . They show that right composition with many series can be performed efficiently. This has for consequence a new algorithm for the efficient conversion of polynomials between various bases including the classical orthogonal polynomials. In the case of general formal orthogonal polynomials, different ideas are required and are developed in [24] .

A characteristic feature of computer algebra systems is their ability to manipulate arbitrarily large integers by storing them over several machine words. By contrast, efficient computations modulo small prime numbers require skillful implementation combining several residues into one machine word. When considering matrices of such residues, several strategies are possible. These are compared from the theoretical and practical point of view by J.-G. Dumas, L. Fousse (Grenoble), and B. Salvy in [20] , [27] . In all cases, the speedup expected in theory, governed by the number of residues that can fit into one word, is reached in practice.

One focus of our project-team is on the complexity analysis of fundamental operations in computer algebra. An operation that had not received sufficient attention is the product of linear differential operators with polynomial coefficients. In [15] , A. Bostan, F. Chyzak, and N. Le Roux present an improved algorithm by reducing the number of matrix products involved. They show that matrix multiplication (in size n× n ) has essentially the same cost as the product of differential operators (of order n and polynomial coefficient degrees n ) when the constants are rational numbers, while the product of operators has an essentially quadratic cost when computations are performed modulo a prime number.

A principal topic of the project-team is the symbolic computation of integrals involving special functions. In this framework, we expect more efficient algorithms provided we focus to more particular classes of inputs. In this spirit, S. Chen has started a PhD thesis in December 2007 under the codirection of F. Chyzak and Z. Li (Chinese Academy of Sciences, Beijing). His work is on the efficiency of the symbolic integration of rational functions, which covers many combinatorial applications.

Structured linear algebra techniques are versatile tools that allow to deal with various types of structured matrices, whether of Toeplitz, Hankel, Vandermonde or Cauchy type. Such linear systems are classically solved by means of a compact representation in operations, where n is the matrix size and is a measure of the structure that can grow with n . A. Bostan, C.-P. Jeannerod (project-team Arenaire), and É Schost showed in [5] that this cost can be reduced to , with <2.38 . The improvement is based on re-introducing fast dense linear algebra for operations on a compact representation of the given matrix. This makes efficient Hermite-Padé approximation and efficient interpolation of bivariate polynomials possible.

The algorithmic advances on power series manipulation over the last decade (computation of high order expansions, Hermite-Padé approximation, ...) have been used by A. Bostan and M. Kauers in tackling a notorious combinatorial conjecture. Their experimental mathematics approach in [23] led to the computer-aided discovery of structural properties of enumerating functions for walks. They have systematically searched for differential and algebraic equations that the series counting the number of walks in the quarter plane satisfy. They have also made a first step towards classifying walks in the first octant of space by considering all step sets with up to five elements, and performed a systematic search for equations of the corresponding series.

M. Mezzarobba's PhD thesis is about basic algorithms operating on D-finite functions, with a special focus on numerical evaluation. With B. Salvy, he developped algorithms to compute tight bounds on P-recursive sequences and D-finite functions. A preliminary implementation of those algorithms in Maple is available in the NumGfun package available from http://algo.inria.fr/mezzarobba/NumGfun-0.2.tgz .

In his master thesis [28] , A. Benoit gave a new algorithm for computing the recurrence of coefficients in a Chebyshev expansion. His algorithm computes the recurrence in a provably and practically more efficient way than previously-known algorithms. He has implemented his algorithm in the computer algebra system Maple.

Part of the activity of the project-team Algorithms is in the project “Dynamic Dictionary of Mathematical Functions” of the Microsoft Research – INRIA Joint Centre. This aims at a new version of the “Encyclopedia of Special Functions” developed in the past by L. Meunier within the project-team ( http://algo.inria.fr/esf ), with the goal of added interactivity. For instance, a user on the web will be able to request more terms of an asymptotic expansion, or improved approximation formulas, which requires incremental symbolic computations. To this end, F. Chyzak is developing a system DynaMoW dedicated to the presentation of interactive mathematics on the web. Its essential feature is to control simultaneously the production of displayed documents and symbolic calculations in possibly concurrent computer-algebra sessions. A prototype is available at http://ddmf.msr-inria.inria.fr . In the future, the “Encyclopedia of Combinatorial Structures”, also a production of the team, will receive the same kind of extensions.