eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2020): 1.045

The field of rational constants of the Volterra derivation; pp. 133–135

Full article in PDF format | doi: 10.3176/proc.2014.2.03

Janusz Zieliński


We describe the field of rational constants of the four-variable Volterra derivation. Thus, we determine all rational first integrals of its corresponding system of differential equations. Such derivations play a role in population biology, laser physics, and plasma physics. Moreover, they play an important part in the derivation theory itself, since they are factorizable derivations. The problem is also linked to the invariant theory.


  1. Almeida, M. A., Magalhães, M. E., and Moreira, I. C. Lie symmetries and invariants of the Lotka–Volterra system. J. Math. Phys., 1995, 36, 1854–1867.

  2. Bogoyavlenskii, O. I. Algebraic constructions of integrable dynamical systems-extension of the Volterra system. Russ. Math. Surv., 1991, 46(3), 1–64.

  3. Cairó, L. and Llibre, J. Darboux integrability for 3D Lotka–Volterra systems. J. Phys. A, 2000, 33, 2395–2406.

  4. Hegedűs, P. The constants of the Volterra derivation. Cent. Eur. J. Math., 2012, 10, 969–973.

  5. Kuroda, S. Fields defined by locally nilpotent derivations and monomials. J. Algebra, 2005, 293, 395–406.

  6. Maciejewski, A. J., Moulin Ollagnier, J., Nowicki, A., and Strelcyn, J.-M. Around Jouanolou non-integrability theorem. Indag. Math. (N.S.), 2000, 11, 239–254.

  7. Nowicki, A. Polynomial Derivations and Their Rings of Constants. N. Copernicus University Press, Toruń, 1994.

  8. Nowicki, A. and Zieliński, J. Rational constants of monomial derivations. J. Algebra, 2006, 302, 387–418.

  9. Ossowski, P. and Zieliński, J. Polynomial algebra of constants of the four variable Lotka–Volterra system. Colloq. Math., 2010, 120, 299–309.

10. Zieliński, J. The five-variable Volterra system. Cent. Eur. J. Math., 2011, 9, 888–896.

11. Zieliński, J. Rings of constants of four-variable Lotka–Volterra systems. Cent. Eur. J. Math., 2013, 11, 1923–1931.

12. Zieliński, J. and Ossowski, P. Rings of constants of generic 4D Lotka–Volterra systems. Czechoslovak Math. J., 2013, 63, 529–538.

Back to Issue