arXiv subjects

Thomas Cluzeau

Publications and source records attributed to Thomas Cluzeau.

6 recordsLinked to original sources

arXiv2026

Schreyer resolution of modules over formal power series

Standard bases of modules over algebras of formal power series play the same role as Gröbner bases of modules over polynomial algebras. In this article, we first prove the analogue of the diamond lemma for modules over formal power series; it characterises standard bases in terms of unique remainders and standard representations. Then, using standard representations, we provide a method to construct a standard basis of the module of syzygies of a standard basis. This construction can be applied inductively to obtain a free resolution, similar to the Schreyer resolution, for finitely presented modules over formal power series.

math.AC↗

arXiv2024

Topological closure of formal power series ideals and application to topological rewriting theory

We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.

math.AC↗

arXiv2020

Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz

Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group.

math.AG↗

arXiv2017

Computing effectively stabilizing controllers for a class of $n$D systems

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimen-sional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc U n = {z = (z 1 ,. .. , z n) $\in$ C n | |z 1 | 1,. .. , |z n | 1}. It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = p 1 ,. .. , p r that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if V (I) = {z $\in$ C n | p 1 (z) = $\times$ $\times$ $\times$ = p r (z) = 0} $\cap$ U n = $\emptyset$. In the present article, we consider the specific class of linear nD systems (which includes the class of 2D systems) for which the ideal I is zero-dimensional, i.e., the p i 's have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V (I) $\cap$ U n = $\emptyset$, as well as for computing, if it exists, a stable polynomial p $\in$ I which allows the effective computation of a stabilizing controller. We illustrate our algorithms through an example and finally provide running times of prototype implementations for 2D and 3D systems.

math.NA↗

arXiv2013

Efficient Algorithms for Computing Rational First Integrals and Darboux Polynomials of Planar Polynomial Vector Fields

We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity $\bigOsoft(N^{2 ω})$ and to a deterministic algorithm solving the problem in $\bigOsoft(d^2N^{2 ω+1})$ arithmetic operations, where $N$ denotes the given bound for the degree of the rational first integral, and where $d \leq N$ is the degree of the vector field, and $ω$ the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in $\bigOsoft(N^{ω+2})$ arithmetic operations. By comparison, the best previous algorithm uses at least $d^{ω+1}\, N^{4ω+4}$ arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package which is available to interested readers with examples showing its efficiency.

cs.SC↗

arXiv2006

Low Complexity Algorithms for Linear Recurrences

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer $N$ (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in $N$. We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree $N$. We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in $O(\sqrt{N}\log^{2}N)$ bit operations; a deterministic one that computes a compact representation of the solution in $O(N\log^{3}N)$ bit operations. Similar speed-ups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.

cs.SC↗