Efficient computation of regular differential systems by change of rankings using Kähler differentials

We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated

Efficient computation of regular differential systems by change of rankings using Kähler differentials

We present two algorithms to compute a regular differential system for some ranking, given an equivalent regular differential system for another ranking. Both make use of Kähler differentials. One of them is a lifting for differential algebra of the FGLM algorithm and relies on normal forms computations of differential polynomials and of Kähler differentials modulo differential relations. Both are implemented in MAPLE V. A straightforward adaptation of FGLM for systems of linear PDE is presented too. Examples are treated

Differential Elimination and Biological Modelling

International audienceThis paper describes applications of a computer algebra method, differential elimination, to applied mathematics problems mostly borrowed from biology. The two considered applications are related to the parameters estimation and the model reduction problems. In both cases, differential elimination can be viewed as a preparation to numerical treatments. Together with the applications, the paper introduces two implementations of the differential elimination algorithms: the diffalg package and the BLAD libraries

A Normal Form Algorithm for Regular Differential Chains

International audienceThis paper presents a new algorithm for computing the normal form of a differential rational fraction modulo differential ideals presented by regular differential chains. An application to the computation of power series solutions is presented and illustrated with the new DifferentialAlgebra MAPLE package

A Differential Algebra Introduction For Tropical Differential Geometry

Doctora

On the role of differential algebra in biological modeling

Extended abstract of an invited talk at Differential Algebra and related Computer Algebra (Catania, Italy, March 28th, 2008)International audienceDifferential algebra is an algebraic theory for studying systems of polynomial ordinary differential equations (ODE). Among all the methods developed for system modeling in cellular biology, it is particularly related to the well-established approach based on nonlinear ODE. A subtheory of the differential algebra, the differential elimination, has proved to be useful in the parameters estimation problem. It seems however still more promising in the quasi-steady state approximation theory, recent results show

A computer scientist point of view on Hilbert's differential theorem of zeros

What is a solution of a system of polynomial differential equations ? This paper provides an original presentation of well-known theorems, with a computer scientist flavor, relying on an improved normal form algorithm

Computing differential characteristic sets by change of ordering

submitted to the Journal of Symbolic ComputationWe describe an algorithm for converting a characteristic set of a prime differential ideal from one ranking into another. This algorithm was implemented in many different languages and has been applied within various software and projects. It permitted to solve formerly unsolved problems

Towards an automated reduction method for polynomial ODE models in cellular biology

International audienceThis paper presents the first version of an algorithmic scheme dedicated to the model reduction problem, in the context of polynomial ODE models derived from generalized chemical reaction systems. This scheme, which relies on computer algebra, is implemented within a new MAPLE package. It is applied over an example. The qualitative analysis of the reduced model is afterwards completely carried out, proving the practical relevance of our methods

Chemical Reaction Systems, Computer Algebra and Systems Biology

International audienceIn this invited paper, we survey some of the results obtained in the computer algebra team of Lille, in the domain of systems biology. So far, we have mostly focused on models (systems of equations) arising from generalized chemical reaction systems. Eight years ago, our team was involved in a joint project, with physicists and biologists, on the modeling problem of the circadian clock of the green algae Ostreococcus tauri. This cooperation led us to different algorithms dedicated to the reduction problem of the deterministic models of chemical reaction systems. More recently, we have been working more tightly with another team of our lab, the BioComputing group, interested by the stochastic dynamics of chemical reaction systems. This cooperation led us to efficient algorithms for building the ODE systems which define the statistical moments associated to these dynamics. Most of these algorithms were implemented in the MAPLE computer algebra software. We have chosen to present them through the corresponding MAPLE packages