François Lemaire

Computing canonical representatives of regular differential ideals

In this paper, we give three theoretical and practical contributions for solving polynomial ODE or PDE systems. The first one is practical: an algorithm which improves the purely algebraic part of Rosenfeld—Gröbner (the polynomial ODE or PDE systems simplifier which is the core of the Maple 5.5 diffalg package). It is a variant of lextriangular but does not need any Gröbner basis computation. The second one is theoretical: a characterization of the output of Rosenfeld—Gröbner and a clarification of the existing relationship between algebraic and differential characteristic sets. The third one is theoretical as well as practical: an algorithm to compute canonical representatives of differential polynomials modulo regular differential ideals without any use of Gröbner bases. This algorithm simplifies the theory (somehow a “pedagogic” contribution) but permits us also to perform easily linear algebra over the base field in the factor differential ring defined by a regular differential ideal.

On proving the absence of oscillations in models of genetic circuits

Using computer algebra methods to prove that a gene regulatory network cannot oscillate appears to be easier than expected. We illustrate this claim with a family of models related to historical examples.

Real Root Isolation of Regular Chains

We present an algorithm RealRootIsolate for isolating the real roots of a polynomial system given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain and a numerical approximation given by intervals. Our algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported.

Applying a Rigorous Quasi-Steady State Approximation Method for Proving the Absence of Oscillations in Models of Genetic Circuits

In this paper, we apply a rigorous quasi-steady state approximation method on a family of models describing a gene regulated by a polymer of its own protein. We study the absence of oscillations for this family of models and prove that Poincare-Andronov-Hopf bifurcations arise if and only if the number of polymerizations is greater than 8. A result presented in a former paper at Algebraic Biology 2007is thereby generalized. The rigorous method is illustrated over the basic enzymatic reaction.

Model Reduction of Chemical Reaction Systems using Elimination

There exist different schemes of model reduction for parametric ordinary differential systems arising from chemical reaction systems. In this paper, we focus on some schemes which rely on quasi-steady states approximations. We show that these schemes can be formulated by means of differential and algebraic elimination. Our formulation is simpler than the classical ones. It permitted us to obtain an approximation of the basic enzymatic reaction system which is different from those of Henri–Michaëlis–Menten and Briggs–Haldane.

Computing differential characteristic sets by change of ordering

We 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.

Oscillations in the expression of a self-repressed gene induced by a slow transcriptional dynamics

We revisit the dynamics of a gene repressed by its own protein in the case where the transcription rate does not adapt instantaneously to protein concentration but is a dynamical variable. We derive analytical criteria for the appearance of sustained oscillations and find that they require degradation mechanisms much less nonlinear than for infinitely fast regulation. Deterministic predictions are confirmed by stochastic simulations of this minimal genetic oscillator.

When does ( T ) equal sat( T )

Given a regular chain T, we aim at finding an efficient way for computing a system of generators of Sat(T), the saturated ideal of T. A natural idea is to test whether the equality {T}=Sat(T) holds, that is, whether T generates its saturated ideal. By generalizing the notion of primitivity from univariate polynomials to regular chains, we establish a necessary and sufficient condition, together with a Grobner basis free algorithm, for testing this equality. Our experimental results illustrate the efficiency of this approach in practice.

Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library

We present new functionalities that we have added to the RegularChains library in Maple to efficiently compute irredundant triangular decompositions. We report on the implementation of different strategies. Our experiments show that, for difficult input systems, the computing time for removing redundant components can be reduced to a small portion of the total time needed for solving these systems.

The RegularChains library in MAPLE

Performing calculations modulo a set of relations is a basictechnique in algebra. For instance, computing the inverse of aninteger modulo a prime integer or computing the inverse of thecomplex number 3 + 2<i>t</i> modulo the relation&ell;² + 1 = 0. Computing modulo a set<i>S</i> containing more than one relation requiresfrom <i>S</i> to have some mathematical structure. Forinstance, computing the inverse of <i>p</i> =<i>x</i> + <i>y</i> modulo<i>S</i> ={<i>x</i>² +<i>y</i> +1,<i>y</i>² +<i>x</i> + 1} is difficult unless one realizes thatthis question is equivalent to computing the inverse of<i>p</i> modulo <i>C</i> ={<i>x</i>⁴ +2<i>x</i>² +<i>x</i> + 2,<i>y</i> +<i>x</i>² + 1}. Indeed, fromthere one can simplify <i>p</i> using<i>y</i> =-<i>x</i>² - 1 leading to<i>q</i> =-<i>x</i>² +<i>x</i> - 1 and compute the inverse of<i>q</i> modulo<i>x</i>⁴ +2<i>x</i>² +<i>x</i> + 2 (using the extended Euclidean algorithm)leading to -1/2<i>x</i>³ -1/2<i>x</i>. One commonly used mathematical structurefor a set of algebraic relations is that of a<i>Gr&ouml;bner basis.</i> It is particularly wellsuited for deciding whether a quantity is null or not modulo a setof relations. For inverse computations, the notion of a<i>regular chain</i> is more adequate. For instance,computing the inverse of <i>p</i> =<i>x</i> + <i>y</i> modulo the set<i>C</i> ={<i>y</i>² -2<i>x</i> +1,<i>x</i>² -3<i>x</i> + 2}, which is both a Gr&ouml;bner basisand a regular chain, is easily answered in this latter point ofview. Indeed, it naturally leads to consider the GCD of<i>p</i> and<i>C<inf>y</inf></i> =<i>y</i>² -2<i>x</i> + 1 modulo the relation<i>C<inf>x</inf></i> =<i>x</i>² -3<i>x</i> + 2 = 0, which is [EQUATION] This shows that <i>p</i> has no inverse if<i>x</i> = 1 and has an inverse (which can be computedand which is -<i>y</i> + 2) if <i>x</i> =2.