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$p$ -adic analysis and classical sequences of numbers. (Analyse $p$ -adique et suites classiques de nombres.)

Barsky, Daniel (1981)

Séminaire Lotharingien de Combinatoire [electronic only]

$p$ -adic difference-difference Lotka-Volterra equation and ultra-discrete limit.

Matsutani, Shigeki (2001)

International Journal of Mathematics and Mathematical Sciences

p-Adic L-Functions for CM Fields.

Nicholas M. Katz (1978)

Inventiones mathematicae

Picard-Vessiot theory in general Galois theory

Hiroshi Umemura (2011)

Banach Center Publications

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

Philippe Robba (1978/1979)

Groupe de travail d'analyse ultramétrique

Polynômes de Barsky

Youssef Haouat, Fulvio Grazzini (1979)

Annales scientifiques de l'Université de Clermont. Mathématiques

Polynomial algebra of constants of the four variable Lotka-Volterra system

Piotr Ossowski, Janusz Zieliński (2010)

Colloquium Mathematicae

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

Jean Moulin Ollagnier, Andrzej Nowicki (1999)

Colloquium Mathematicae

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 $d = x (C y + z) \frac{\partial}{\partial x} + y (A z + x) \frac{\partial}{\partial y} + z (B x + y) \frac{\partial}{\partial z}$ , called the Lotka-Volterra derivation, where A,B,C ∈ k.

Polynomial Riccati equations with algebraic solutions

Henryk Żołądek (2002)

Banach Center Publications

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.