Fields of rational constants of cyclic factorizable derivations

Abstract

We describe all rational constants of a large family of four-variable cyclic factorizable derivations. Thus, we determine all rational first integrals of their corresponding systems of differential equations. Moreover, we give a characteristic of all four-variable Lotka-Volterra derivations with a nontrivial rational constant. All considerations are over an arbitrary field of characteristic zero. Our main tool is the investigation of the cofactors of strict Darboux polynomials. Factorizable derivations are important in derivation theory. Namely, we may associate the factorizable derivation with any given derivation of a polynomial ring and that construction helps to determine rational constants of arbitrary derivations. Besides, Lotka-Volterra systems play a significant role in population biology, laser physics and plasma physics.

Keywords

• Lotka-Volterra derivation
• factorizable derivation
• rational constant
• rational first integral