-adic difference-difference Lotka-Volterra equation and ultra-discrete limit.
p-Adic L-Functions for CM Fields.
Picard-Vessiot theory in general Galois theory
We give a transparent proof that difference Picard-Vessiot theory is a part of the general difference Galois theory. We apply the proof to iterative q-difference Picard-Vessiot theory to show that Picard-Vessiot theory for iterative q-difference field extensions is in the scope of the general Galois theory of Heiderich. We also show that Picard-Vessiot theory is commutative in the sense that studying linear difference-differential equations, no matter how twisted the operators are, we cannot encounter...
Points singuliers d'équations différentielles
Polynômes de Barsky
Polynomial algebra of constants of the four variable Lotka-Volterra system
We describe the ring of constants of a specific four variable Lotka-Volterra derivation. We investigate the existence of polynomial constants in the other cases of Lotka-Volterra derivations, also in n variables.
Polynomial algebra of constants of the Lotka-Volterra system
Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form , called the Lotka-Volterra derivation, where A,B,C ∈ k.
Polynomial Riccati equations with algebraic solutions
We consider the equations of the form dy/dx = y²-P(x) where P are polynomials. We characterize the possible algebraic solutions and the class of equations having such solutions. We present formulas for first integrals of rational Riccati equations with an algebraic solution. We also present a relation between the problem of algebraic solutions and the theory of random matrices.
Power Series Solutions of Algebraic Differential Equations.
Powersum formula for differential resolvents.
Powersum formula for polynomials whose distinct roots are differentially independent over constants.
Primitives des fonctions élémentaires
Produit symmétrique de l'équation de Bessel
Prolongements des éléments algébriques
Proof of the combinatorial nullstellensatz over integral domains, in the spirit of Kouba.
Propriétés algébriques de suites différentiellement finies
Propriétés algébriques des opérateurs d'Airy de petit ordre
Puiseux series solutions of ordinary polynomial differential equations: complexity study.
Quelques remarques sur les opérateurs différentiels d'ordre 1
Rational Constants of Generic LV Derivations and of Monomial Derivations
We describe the fields of rational constants of generic four-variable Lotka-Volterra derivations. Thus, we determine all rational first integrals of the corresponding systems of differential equations. Such systems play a role in population biology, laser physics and plasma physics. They are also an important part of derivation theory, since they are factorizable derivations. Moreover, we determine the fields of rational constants of a class of monomial derivations.