Evelyne Hubert

Publications

• Generation properties of Maurer-Cartan invariants Preprint [hal:inria-00194528]
  
  Abstract: For the action of a Lie group, which can be given by its infinitesimal generators only, we characterize a generating set of differential invariants of bounded cardinality and show how to rewrite any other differential invariants in terms of them. Those invariants carry geometrical significance and have been used in equivalence problem in differential geometry.
  Keywords: Lie group actions; Differential invariants; Moving frame; Maurer-Cartan forms.
  
  
• Differential invariants of a Lie group action: syzygies on a generating set. Journal of Symbolic Computation, 44:4 pages 382-416 (2009). [doi:10.1016/j.jsc.2008.08.003]
  
  Abstract: Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential invariants.
  Keywords: Lie group actions;x Differential invariants; Moving frame; Syzygies; Differential algebra; Symbolic computation
  
  
• Probabilistic Algorithms for Computing Resolvent Representations of Regular Differential Ideals; with T. Cluzeau. Applicable Algebra and Error Correcting Codes, volume 19, number 5, p 365-392 [SpringerLink]. Preprint [hal:inria-00089287].
  
  Abstract: In a previous article, we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations. We propose two different approaches. The first one uses differential characteristic decompositions whereas the second one proceeds by prolongation and algebraic elimination. Both constructions depend on the choice of a tuple over the differential base field and their success relies on the chosen tuple to be separating. The probabilistic aspect of the algorithms comes from this choice. To control it, we exhibit a family of tuples for which we can bound the probability that one of its element is separating.
  
  
• Differential Invariants of Conformal and Projective Surfaces; with P. J. Olver, in Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, Symmetry, Integrability and Geometry: Methods and Applications, volume 3 (2007). [SIGMA 2007-097]
  
  Abstract: We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
  Keywords: conformal differential geometry; projective differential geometry; differential invariants; moving frame; syzygy; differential algebra; symbolic computation.
  
  
• Smooth and Algebraic Invariants of a Group Action. Local and Global Constructions; with I. Kogan. Foundations Computational Mathematics. 7:4 pages 345-383 (2007) [Springer Link]
  
  Abstract: Motivated by a wealth of applications both the theory of differential invariants and the theory of algebraic invariants have versed in computational mathematics. Differential invariants are intimately linked with studying differential systems, while algebraic theories give proper grounds to symbolic algorithms. The ambition of this paper is to provide algebraic foundations to the moving frame method for the construction of local invariants and present a parallel algebraic construction that produces algebraic invariants together with the relations they satisfy. We then show that the algebraic setting offers a computational solution to the differential geometric construction.
  Keywords: rational and algebraic invariants; smooth and differential invariants; algebraic and Lie group actions; cross-section; moving frame method; Groebner basis.
  
  
• Rational Invariants of a Group Action. Construction and Rewriting; with I. Kogan. Journal of Symbolic Computation, 42:1-2 pages 203-217 (2007). [ScienceDirect]
  
  Attachment: Maple code applied to the examples treated in the paper.
  
  Abstract: Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gr\"obner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zero-dimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gr\"obner basis allows to express any rational invariant in terms of the generators.
  Keywords: Algebraic Invariants, Moving Frame, Groebner Bases, Computational Algebra.
  
  
• Rational and Replacement Invariants of a Group Action; with I. Kogan. ACM SIGSAM Bulletin, Volume 39, Issue 3 (2005). [Award winning poster]
  
  
  
• Improvements to a triangulation-decomposition algorithm for ordinary differential systems in higher degree cases. ISSAC 2004. [ACM portal]
  
  Attachment: Experimental evaluation of performances
  
  Abstract: We introduce new ideas to improve the efficiency and rationality of a triangulation decomposition algorithm. On the one hand we identify and isolate the polynomial remainder sequences in the triangulation-decomposition algorithm. Subresultant polynomial remainder sequences are then used to compute them and their specialisation properties are applied for the splittings. The gain is two fold: control of expression swell and reduction of the number of splittings. On the other hand, we remove the role that initials had in previous triangulation-decomposition algorithms. They are not needed in theoretical results and it was expected that they need not appear in the algorithms. This is the case of the algorithm presented.
  Keywords: systems of differential equations, differential elimination, differential ideal theory, triangular sets, subresultant polynomial remainder sequence.
  
  
• Differential Algebra for Derivations with Nontrivial Commutation Rules, Journal of Pure and Applied Algebra. Volume 200, Issues 1-2, 1 August 2005, Pages 163-190. [ Science Direct ].
  
  Abstract: The classical assumption of differential algebra, differential elimination theory and formal integrability theory is that the derivations do commute. That is the standard case arising from systems of partial differential equations written in terms of the derivations w.r.t. the independant variables. We inspect here the case where the derivations satisfy nontrivial commutation rules. That situation arises for instance when we consider a system of equations on the differential invariants of a Lie group action. We develop the algebraic foundations for such a situation. They lead to algorithms for completion to formal integrability and differential elimination.
  Keywords: Differential Algebra - Differential Elimination - Differential Invariants
  
  
• Computing Power Series Solutions of Nonlinear PDE systems, with N. Le Roux, ISSAC 2003. [ ACM portal]
  
  Abstract: This paper presents a new algorithm to compute the power series solutions of a system of a significant class of nonlinear systems of partial differential equations. The algorithm presented is very different from previous algorithms to perform this task. Those relie on differentiating iteratively the differential equations to get coefficients of the power series one at a time. The here presented algorithm relies on using the linearisation of the system and is thus in the line of Newton methods. At each step the order up to which the power series solution is known is doubled.
  Keywords: nonlinear partial differential systems - power series solutions - differential algebra - formal integrability.
  
  
• Notes on triangular sets and triangulation-decomposition algorithms I: Polynomial systems. Chapter of Symbolic and Numerical Scientific Computations Edited by U. Langer and F. Winkler. LNCS, volume 2630, Springer-Verlag Heidelberg. [Springer Link ] ( indexed pdf version)
  
  Abstract: This is the first in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.%, most of which are not original. We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. We address here polynomial systems and some of the material we develop here will be used in the second part, devoted to differential systems.
  Keywords: polynomial systems, triangular sets, regular chains, characteristic sets, algorithms.
  
  
• Notes on triangular sets and triangulation-decomposition algorithms II: Differential Systems. Chapter of Symbolic and Numerical Scientific Computations Edited by U. Langer and F. Winkler. LNCS, volume 2630, Springer-Verlag Heidelberg. [Springer Link ] ( indexed pdf version)
  
  Attachment: How to treat the examples of applications given in the paper with the Maple library diffalg.
  
  Abstract: This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results. We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.
  Keywords: systems of nonlinear partial differential equations, differential algebra, differential ideal theory, algorithms.
  
  
• Resolvent Representation for Regular Differential Ideals, with T. Cluzeau, Applicable Algebra in Engineering, Communication and Computing, 13:5, pages 395-425, 2003.
  
  Abstract: We show that the generic zeros of a differential ideal $[A]:H_A^\infty$ defined by a differential chain $A$ are birationally equivalent to the general zeros of a single regular differential polynomial. This provides a generalization of both the cyclic vector construction for system of linear differential equations and the rational univariate representation of algebraic zero dimensional radical ideals.
  Keywords: differential algebra, differential primitive element, cyclic vector, computer algebra, resolvent.
  
  
• Factorization free decomposition algorithms in differential algebra, Journal of Symbolic Computations, 2000, volume 29(4-5) pp641-662.
  
  Abstract: Insight on the structure of differential ideals defined by coherent autoreduced set allows one to uncouple the differential and algebraic computations in a decomposition algorithm. Original results as well as concise new proofs of already presented theorems are exposed. As a consequence, an effective version of Ritt's algorithm can be simply described.
  
  
• A note on the Lax pairs for Painlevé equations, with A. Kapaev, Journal of Physics. A. Mathematical and General, volume 32, number 46 (1999).
  
  Abstract: The theory of integrable integral operators suggests a new way for generating Lax pair for the classical Painelvé equations. This method involves a gauge transformation applied to a linear system exactly solvable in terms of the classical special funstions. Some of the Lax pairs we introduce are known, others are new. The computationnal investigations were lead with diffalg(99), a MapleV package for differential elimination.
  
  
• Essential components of an algebraic differential equation , Journal of Symbolic Computations, 1999, volume 28(4-5), 657-680.
  
  Abstract: We present an algorithm to determine the essential singular components of an algebraic differential equation. Geometrically, this corresponds to determine the singular solutions that have enveloping properties. The algorithm is practical and efficient because it is factorisation free, unlike the previous such algorithm.
  This paper is self contained in the sense that it requires nearly only to textbooks (Mainly Kolchin, 1973, Differential Algebra and Algebraic Groups).
  
  
• Constructing stable manifolds of stationary solutions, with G. Moore, IMA J. Numer. Anal., 1999, volume 19(3), pp 375-424.
  
  Comment: See more links on the subject on the web site of my MSc advisor Pr. G. Moore.
  
  
• Detecting Degenerate Cases in Non-Linear Differential Equations of First order, Theoretical Computer Science,1997, vol 187 (1-2), pages 7-25.
  
  Abstract: Differential geometric and algebraic ideas are exposed and transposed into algorithms to detect and classify degenerate behaviors appearing in first order ordinary differential equations. Differential geometry is considered for a local analysis. It indeed gives insight on the behavior of the solutions at singular points. A sequence of algebraic operations will be put forth to split the locus of singular points according to their properties. When part of the locus of singular points turns out to be a singular solution, the set of solutions splits and shall be considered from a global standpoint. To this aim, differential algebra allows to define rigorously the general and particular solutions. Investigating further leads to an original algorithm to compute a differential basis of the general solution. This basis give means to determine which singular solutions are particular solutions.
  Keywords: Singularities of Ordinary Differential Equations - Singular Solutions - Particular Solutions - General Solution - Differential Algebra.
  
  
• The general solution of an ordinary differential equation ISSAC 1996.[ACM portal]
  
  Keywords: General Solution, Singular Solutions, Differential algebra, Formal Power Series Solution.
  
  
• Algebra and Algorithms for Implicit Differential Equations, PhD thesis, Institut National Polytechnique de Grenoble April 1997. [Postcript files].
  
  Abstract: The set of solutions of an algebraic differential equation splits into the general solution and the singular solutions. These two concepts are given a precise meaning in the setting of differential algebra, a theory initiated by J.F.Ritt. Recent progresses in that field have brought effective algorithms that decide of the triviality of a differential system and provide a first decompo sition of the set of solutions. We can thus determine if a differential equation admits any singular solutions and describe those. The decompositions there obtained are nonetheless not minimal. In this memoir, we propose a factorization free algorithm that eliminates the redundant components. The analytic interpretation is that we split the set of singular solutions into the singular solutions that are envelopes of the general solution and the singular solutions which are the limits of some other solutions. The crux of the algorithm lies in the Low power theorem, while the effective implementation is based on the Rosenfeld-Gröbner algorithm. We furthermore present an algorithm and some criteria to compute the differential bases of the essential components of a differential equation. Such bases allow an analysis of singularities and integration heuristics.
  
  
• Computation of Stable Manifolds, Master Report, G. Moore advisor, September 1993. [postcript]
  
  Abstract: Direct methods are presented to compute the two dimensional stable or unstable manifold of a saddle fixed point in a third order dynamical sytem. The algorithms are based on the invariance property of the manifold and the choice of a parametrization. Two of them are considered. The first one is a cylindrical coordinate system based on the tangent space, the second a geodesic polar coordinate system. Both formulations lead to a partial differential equation of first order with $2 \pi$ periodic boundary conditions. A spectral method based on a Fourier system is thus suitable to compute a numerical solution.
  Keywords: Invariant manifolds of dynamical sytems - Spectral methods.

HOME PAGE