Polynomial algebra of constants of the Lotka-Volterra system

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Authors ¹, ²

GAGE, UMS CNRS 658 Medicis, École Polytechnique, F-91128 Palaiseau Cedex, France
Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

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.

B. Grammaticos, J. Moulin Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in $ℝ^{3}$ : the Lotka-Volterra system, Phys. A 163 (1990), 683-722.
J. Moulin Ollagnier, Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math. 121 (1997), 463-476.
J. Moulin Ollagnier, Rational integration of the Lotka-Volterra system, ibid. 123 (1999), 437-466.
J. Moulin Ollagnier, A. Nowicki and J.-M. Strelcyn, On the non-existence of constants of derivations: the proof of a theorem of Jouanolou and its development, ibid. 119 (1995), 195-233.
A. Nowicki, Polynomial derivations and their rings of constants, N. Copernicus University Press, Toruń, 1994.
A. Nowicki, On the non-existence of rational first integrals for systems of linear differential equations, Linear Algebra Appl. 235 (1996), 107-120.
A. Nowicki and M. Nagata, Rings of constants for k-derivations in $k [x_{1}, ..., x_{n}]$ , J. Math. Kyoto Univ. 28 (1988), 111-118.
A. Nowicki and J.-M. Strelcyn, Generators of rings of constants for some diagonal derivations in polynomial rings, J. Pure Appl. Algebra 101 (1995), 207-212.