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Let k be a field of chracteristic p > 0. We describe all derivations of the polynomial algebra k[x,y], homogeneous with respect to a given weight vector, in particular all monomial derivations, with the ring of constants of the form $k [x^{p}, y^{p}, f]$ , where $f \in k [x, y] ∖ k [x^{p}, y^{p}]$ .

On Second Order Linear Differential Equations with Few Transcendental Solutions.

Hans Peter Rehm (1986)

Manuscripta mathematica

On the bialgebra of functional graphs and differential algebras.

Ginocchio, Maurice (1997)

Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]

On the calculation of some differential galois groups.

N.M. Katz (1987)

Inventiones mathematicae

On the Galois theory of difference fields.

Robert P. Infante (1981)

Aequationes mathematicae

On the geometrization of a lemma of Singer and van der Put

Colas Bardavid (2011)

Banach Center Publications

In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory, used by Singer and van der Put in their reference book. This geometrization, in addition of giving a nice insight on this result, offers us the opportunity to investigate several points of differential algebra and differential algebraic geometry. We study the class of simple Δ-schemes and prove that they all have a coarse space of leaves. Furthermore, instead of considering schemes endowed with...

On the problem of Baer and Kolchin in the Picard-Vessiot theory

Beata Kocel-Cynk, Elżbieta Sowa (2011)

Banach Center Publications

We present the history of the development of Picard-Vessiot theory for linear ordinary differential equations. We are especially concerned with the condition of not adding new constants, pointed out by R. Baer. We comment on Kolchin's condition of algebraic closedness of the subfield of constants of the given differential field over which the equation is defined. Some new results concerning existence of a Picard-Vessiot extension for a homogeneous linear ordinary differential equation defined over...

On the structure of linear recurrent error-control codes

Michel Fliess (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.

On the structure of linear recurrent error-control codes

Michel Fliess (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Opérateurs hypergéométriques réductibles : décompositions et groupes de Galois différentiels

Katy Boussel (1996)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Opération de Cartier et vecteurs de Witt

Gilles Christol (1970/1971)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Ordinary linear differential equations - a survey of the global theory

Neuman, František (1986)

Equadiff 6

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form $H = 1 / 2 \sum_{i = 1}^{n} p ²_{i} + V (q)$ , where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form $V = V_{k} + V_{K}$ , where $V_{k}$ and $V_{K}$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable...

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