Polynomial algebra of constants of the Lotka-Volterra system

• Autorzy: Jean Moulin Ollagnier , Andrzej Nowicki
• Języki publikacji: EN

Abstrakty

EN

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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1999
• Tom: 81
• Numer: 2
• Strony: 263-270

Daty

wydano

1999

otrzymano

1998-11-12

poprawiono

1999-02-24

Twórcy

autor

Jean Moulin Ollagnier

• GAGE, UMS CNRS 658 Medicis, École Polytechnique, F-91128 Palaiseau Cedex, France

autor

Andrzej Nowicki

• Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [1] 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.
• [2] J. Moulin Ollagnier, Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math. 121 (1997), 463-476.
• [3] J. Moulin Ollagnier, Rational integration of the Lotka-Volterra system, ibid. 123 (1999), 437-466.
• [4] 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.
• [5] A. Nowicki, Polynomial derivations and their rings of constants, N. Copernicus University Press, Toruń, 1994.
• [6] A. Nowicki, On the non-existence of rational first integrals for systems of linear differential equations, Linear Algebra Appl. 235 (1996), 107-120.
• [7] 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.
• [8] 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.