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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...

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 ) ∂ ∂ x + y ( A z + x ) ∂ ∂ y + z ( B x + y ) ∂ ∂ 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.

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