Section: New Results

Algebraic computing and high-performance kernels

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. In [18], we extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.

Hermite-Padé approximant bases

In [46] we design fast algorithms for the computation of approximant bases in shifted Popov normal form. For $𝖪$ a commutative field, let $F$ be a matrix in $𝖪{\left[x\right]}^{m\times n}$ (truncated power series) and $\overrightarrow{d}$ be a degree vector, the problem is to compute a basis $P\in 𝖪{\left[x\right]}^{m\times m}$ of the $𝖪\left[x\right]$-module of the relations $p\in 𝖪{\left[x\right]}^{1\times m}$ such that $p\left(x\right)·F\left(x\right)\equiv 0mod{x}^{\overrightarrow{d}}$. We obtain improved complexity bounds for handling arbitrary (possibly highly unbalanced) vectors $\overrightarrow{d}$. We also improve upon previously known algorithms for computing $P$ in normalized shifted form for an arbitrary shift. Our approach combines a recent divide and conquer strategy which reduces the general case to the case where information on the output degree is available, and partial linearizations of the involved matrices.

Resultant of bivariate polynomials

We have proposed in [42] an algorithm for computing the resultant of two generic bivariate polynomials over a field $𝖪$. For such $p$ and $q$ in $𝖪[x,y]$ both of degree $d$ in $x$ and $n$ in $y$, the algorithm computes the resultant with respect to $y$ using ${\left(n^{2-1/\omega }d\right)}^{1+o\left(1\right)}$ arithmetic operations, where $\omega$ is the exponent of matrix multiplication. Previous algorithms from the early 1970's required time ${\left(n^{2}d\right)}^{1+o\left(1\right)}$. We have also described some extensions of the approach to the computation of generic Gröbner bases and of characteristic polynomials of generic structured matrices and in univariate quotient algebras.

Recursive Combinatorial Structures: Enumeration, Probabilistic Analysis and Random Generation

The probabilistic behaviour of many data-structures, like series-parallel graphs used as a running example is this tutorial [13], can be analysed very precisely, thanks to a set of high-level tools provided by Analytic Combinatorics, as described in the book by Flajolet and Sedgewick. In this framework, recursive combinatorial definitions lead to generating function equations from which efficient algorithms can be designed for enumeration, random generation and, to some extent, asymptotic analysis. With a focus on random generation, this tutorial given at STACS first covers the basics of Analytic Combinatorics and then describes the idea of Boltzmann sampling and its realisation. The tutorial addresses a broad TCS audience and no particular pre-knowledge on analytic combinatorics is expected.

Linear Differential Equations as a Data-Structure

A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be computed. A variety of algorithms has thus been designed in recent years that do not aim at “solving”, but at computing with this representation. Many of these results are surveyed in [11].