We introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of finitely generated difference-differential field extensions. Furthermore, we find new invariants of such extensions and show how the computation of multivariate difference-differential polynomials is applied to the equivalence problem for systems of algebraic difference-differential equations.

Multivariate difference-differential dimension polynomials and new invariants of difference-differential field extensions

Les travaux presentes dans ce memoire se basent sur les apports de l'algebre differentielle et les methodes du calcul symbolique pour resoudre des problemes d'automatique non lineaire qui ne se pretent pas a une resolution numerique directe. Le probleme de l'observabilite algebrique locale consiste a decider si les variables d'etat intervenant dans un modele peuvent etre determinees en fonction des entrees et des sorties supposees parfaitement connues. Nous presentons un algorithme probabiliste de complexite arithmetique polynomiale en la taille de l'entree permettant de tester l'observabilite algebrique locale en determinant les variables non observables. L'utilisation du calcul modulaire permet d'obtenir pour ce test une complexite binaire elle aussi polynomiale. Cette complexite depend lineairement de la probabilite de succes qui peut etre arbitrairement fixee. Une implantation de cet algorithme permet de traiter des problemes inaccessibles jusqu'a present. A partir de ces methodes melant calcul symbolique et calcul numerique, nous proposons une generalisation de la notion de platitude differentielle a certains modeles non lineaires decrits par des equations aux derivees partielles. Un systeme differentiel ordinaire est differentiellement plat si ses solutions peuvent etre localement parametrees bijectivement par des fonctions arbitraires. Pour etudier certains systemes d'equations aux derivees partielles non lineaires, on se ramene a un systeme d'equations differentielles ordinaires par discretisation ; notre approche consiste a chercher des discretisations plates telles que les parametrages associes convergent lorsque le pas de discretisation tend vers zero. Cette methode est illustree par l'etude du probleme de planification de trajectoire realisee pour trois modeles non lineaires de dimension infinie : l'equation de la chaleur semilineaire, l'equation de Burger avec diffusion et un modele non lineaire de tige flexible.

https://pastel.archives-ouvertes.fr/tel-00401888/document

Méthodes seminumériques en algèbre différentielle; applications à l'étude des propriétés structurelles de systèmes différentiels algébriques en automatique

pdf/a-geometric-index-reduction-method-for-implicit-systems-of-29ham4yrub.pdf

A geometric index reduction method for implicit systems of differential algebraic equations

In this paper we prove Jacobi’s bound for systems of n partial differential polynomials in n differential variables, which are independent over a prime differential ideal . This generalizes on the one hand our result (Kondratieva et al. in On Jacobi’s bound for systems of differential polynomials, algebra. Moscow University Press, Moscow, pp 79–85, 1982) about Jacobi’s bound for ordinary differential polynomials independent over a prime differential ideal and, on the other hand, the result by Tomasovic (A generalized Jacobi conjecture for arbitrary systems of algebraic differential equations. PhD thesis, Columbia University, 1976), who proposed a version of Jacobi’s bound for partial differential polynomials and proved it in the linear case. In Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, 1999) we gave another proof of this result by Tomasovic. The exposition in the present paper follows our proof in Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, pp 273–280, 1999).

Jacobi’s bound for independent systems of algebraic partial differential equations

In this paper we present Maple implementations of two algorithms developed by M. Zhou and F. Winkler for computing a relative Grobner basis of a finitely generated difference-differential module and for c omputing the bivariate difference-differential dimension polyomial of the module with respect to the natural bifiltration of the ring of difference-differen tial operators. The notion of relative Grobner basis and its use for computing bivariate difference-differential dimension polynomials is explaine d. After this the implementations of the two algorithms are illustrated by a couple of examples.

ial Dimension Polynomials and Their Computation in Maple

Let L and K be differential fields with L an extension of K. It is shown how the module of Kahler differentials S41/K can be used to "linearize" properties of a differential field extension L/K. This is done without restriction on the characteristic p and yields a theory which for p # 0 is no harder than the case p = 0. As an application a new proof of the Ritt basis theorem is given. Introduction. Let k be a field and A a finitely generated algebra over k. One can study the properties of A by taking a surjective homomorphism of k-algebras 4: k[y1, ... ,y.] -> A where Yi ... , y,n are indeterminates over k and then analyzing the kernel I of 4. This is for instance done in [1] for the case of A an integral domain. There one assumes that 4 is chosen so that 0(Yi), ..., 4)(yp) is a transcendance basis for the quotient field of A over k. One then chooses nonzero Fj in k[y1,...,yj] nfI and dj in N for p < j < n such that (1) dj is the degree of Fj in yj, (2) Degreeyi Fj < di if p < i < j, (3) dj is as small as possible. One then observes that if Sj is defined by Fj = Sjy4i + terms of degree less than dj inyj and if S = Sp+, ... Sn, then S 4 I and k[y1, .... ,Yn, I/S]I is generated as an ideal of k[y, . .. ,Yn, 1/S] by p+1,~ .. * * Fn. The subject of this paper is the study of a finitely generated differential algebra over a differential field. This matter has of course been considered before. However to prove anything beyond the most trivial statements one has customarily generalized the above-mentioned procedure, and this has led ultimately to a fairly complicated theory. Such theory is more complicated in nonzero characteristic than in characteristic zero. The purpose of this paper is to show that we may achieve our results in another way, and so that procedure is not followed here. Instead differential algebras are studied by reducing the number of derivation operators involved one at a time. To illustrate this method I have included a new proof of Seidenberg's version of the Ritt-Raudenbusch basis theorem. The primary tool used in reducing the number of derivation operators is the theory of Kiihler differentials, a method for "linearizing" much of the ring theory Received by the editors June 14, 1972 and, in revised form, December 15, 1972. AMS (MOS) subject classifications (1970). Primary 12H05.

/pdf/kahler-differentials-and-differential-algebra-in-arbitrary-1gxx8irns5.pdf

Kähler differentials and differential algebra in arbitrary characteristic

/pdf/systems-of-n-partial-differential-equations-in-n-unknown-3whcedq96d.pdf

Systems of $n$ partial differential equations in $n$ unknown functions: the conjecture of M. Janet

Let D be a linear partial differential operator between vector bundles on a differentiable manifold X of dimension n. Let D be the sheaf of germs of differentiable functions on X. For every h E Z a spectral sequence (hEPq) is associated to D. When D satisfies appropriate regularity conditions these spectral sequences degenerate for all sufficiently large h and hE2?o is the pth Spencer cohomology for D. One can compute hE2 as the cohomology at APT* C, Rh -p.q of a complex 0 --> Rhq * * Al T* (D0, Rh -i1q --> An T'( 0Rh n,q ? 0. When q=0 this complex coincides with the usual (first) Spencer complex for D. These results give a generalization of Spencer's theory. The principal importance of this generalization is that it greatly clarifies the role played by homological algebra in the theory of overdetermined systems of linear partial differential equations.

/pdf/on-spencer-s-cohomology-theory-for-linear-partial-1skryf7x0t.pdf

On Spencer’s cohomology theory for linear partial differential operators

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Papers

Kähler differentials and differential algebra in arbitrary characteristic

Systems of $n$ partial differential equations in $n$ unknown functions: the conjecture of M. Janet

On Spencer’s cohomology theory for linear partial differential operators