Romain Lebreton

On the complexity of solving bivariate systems: the case of non-singular solutions

We give an algorithm for solving bivariate polynomial systems over either <i>k</i>(<i>T</i>)[<i>X,Y</i>] or <i>Q</i>[<i>X,Y</i>] using a combination of lifting and modular composition techniques.

Relaxed p-adic Hensel lifting for algebraic systems

In a previous article [1], an implementation of lazy <i>p</i>-adic integers with a multiplication of quasi-linear complexity, the so-called relaxed product, was presented. Given a ring <i>R</i> and an element <i>p</i> in <i>R</i>, we design a relaxed Hensel lifting for algebraic systems from <i>R/</i> (<i>p</i>) to the <i>p</i>-adic completion <i>R</i>_<i>p</i> of <i>R</i>. Thus, any root of linear and algebraic regular systems can be lifted with a quasi-optimal complexity. We report our implementations in C++ within the computer algebra system Mathemagix and compare them with Newton operator. As an application, we solve linear systems over the integers and compare the running times with Linbox and IML.

Power series solutions of singular (q)-differential equations

We provide algorithms computing power series solutions of a large class of differential or q-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.

Algorithms for the universal decomposition algebra

Let <i>k</i> be a field and let <i>f</i> ∈ <i>k</i> [<i>T</i>] be a polynomial of degree <i>n</i>. The <i>universal decomposition algebra</i> A is the quotient of <i>k</i> [<i>X</i>₁,...,<i>X</i>_<i>n</i>] by the ideal of <i>symmetric relations</i> (those polynomials that vanish on all permutations of the roots of <i>f</i>). We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, <i>i.e</i>. an isomorphism of the form A <i>k</i>[<i>T</i>]/<i>Q</i>(<i>T</i>), since in this representation, arithmetic operations in A are known to be quasi-optimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A.

Algorithms for Structured Linear Systems Solving and Their Implementation

There exists a vast literature dedicated to algorithms for structured matrices, but relatively few descriptions of actual implementations and their practical performance in symbolic computation. In this paper, we consider the problem of solving Cauchy-like systems, and its application to mosaic Toeplitz systems, in two contexts: first in the unit cost model (which is a good model for computations over finite fields), then over Q. We introduce new variants of previous algorithms and describe an implementation of these techniques and its practical behavior. We pay a special attention to particular cases such as the computation of algebraic approximants.

A simple and fast online power series multiplication and its analysis

This paper focuses on online (or relaxed) algorithms for the multiplication of power series over a field and their complexity analysis. We propose a new online algorithm for the multiplication using middle and short products of polynomials as building blocks, and we give the first precise analysis of the arithmetic complexity of various online multiplications. Our algorithm is faster than Fischer and Stockmeyers by a constant factor; this is confirmed by experimental results.

Structured FFT and TFT: symmetric and lattice polynomials

In this paper, we consider the problem of efficient computations with structured polynomials. We provide complexity results for computing Fourier Transform and Truncated Fourier Transform of symmetric polynomials, and for multiplying polynomials supported on a lattice.

Simultaneous Conversions with the Residue Number System Using Linear Algebra

We present an algorithm for simultaneous conversions between a given set of integers and their Residue Number System representations based on linear algebra. We provide a highly optimized implementation of the algorithm that exploits the computational features of modern processors. The main application of our algorithm is matrix multiplication over integers. Our speed-up of the conversions to and from the Residue Number System significantly improves the overall running time of matrix multiplication.

On the complexity of computing certain resultants

Computing resultants is a fundamental algorithmic question, at the heart of higher-level algorithms for solving systems of equations, computational topology, etc. However, in many situations, the best known algorithms are still sub-optimal. The following table summarizes the best results known to us (from [3]), using soft-Oh notation to omit logarithmic factors. In all cases, we assume that f, g have coefficients in a field k, and that their partial degrees in all variables is at most d. The partial degree in all remaining variables of their resultant r = res(f, g, x1) is then at most 2d 2. In this note, the cost of an algorithm is the number of arithmetic operations in k it performs.

Root lifting techniques and applications to list decoding

Motivatived by Guruswami and Rudras construction of folded Reed-Solomon codes, we give algorithms to solve functional equations of the form Q(x, f(x), f(x)) = 0, where Q is a trivariate polynomial. We compare two approaches, one based on Newtons iteration and the second using relaxed series techniques.