Geodesic
Rings of constants of generic 4D Lotka-Volterra systems
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 529-538
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We show that the rings of constants of generic four-variable Lotka-Volterra derivations are finitely generated polynomial rings. We explicitly determine these rings, and we give a description of all polynomial first integrals of their corresponding systems of differential equations. Besides, we characterize cofactors of Darboux polynomials of arbitrary four-variable Lotka-Volterra systems. These cofactors are linear forms with coefficients in the set of nonnegative integers. Lotka-Volterra systems have various applications in such branches of science as population biology and plasma physics, among many others.
We show that the rings of constants of generic four-variable Lotka-Volterra derivations are finitely generated polynomial rings. We explicitly determine these rings, and we give a description of all polynomial first integrals of their corresponding systems of differential equations. Besides, we characterize cofactors of Darboux polynomials of arbitrary four-variable Lotka-Volterra systems. These cofactors are linear forms with coefficients in the set of nonnegative integers. Lotka-Volterra systems have various applications in such branches of science as population biology and plasma physics, among many others.
DOI : 10.1007/s10587-013-0035-z
Classification : 12H05, 13N15, 34A34, 92D25
Keywords: Lotka-Volterra derivation; polynomial constant; polynomial first integral; Darboux polynomial 
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     journal = {Czechoslovak Mathematical Journal},
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Zieliński, Janusz; Ossowski, Piotr. Rings of constants of generic 4D Lotka-Volterra systems. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 529-538. doi: 10.1007/s10587-013-0035-z

[1] Bogoyavlenskij, O. I.: Algebraic constructions of integrable dynamical systems: extension of the Volterra system. Russ. Math. Surv. 46 (1991), 1-64 translation from Usp. Mat. Nauk 46 (1991), 3-48 Russian. | MR

[2] Jędrzejewicz, P.: Positive characteristic analogs of closed polynomials. Cent. Eur. J. Math. 9 (2011), 50-56. | DOI | MR | Zbl

[3] Kuroda, S.: Fields defined by locally nilpotent derivations and monomials. J. Algebra 293 (2005), 395-406. | DOI | MR | Zbl

[4] Ollagnier, J. Moulin: Polynomial first integrals of the Lotka-Volterra system. Bull. Sci. Math. 121 (1997), 463-476. | MR

[5] Moulin-Ollagnier, J., Nowicki, A.: Polynomial algebra of constants of the Lotka-Volterra system. Colloq. Math. 81 (1999), 263-270. | DOI | MR | Zbl

[6] Nowicki, A.: Polynomial Derivations and Their Rings of Constants. Uniwersytet Mikołaja Kopernika, Toruń (1994). | MR | Zbl

[7] Nowicki, A., Nagata, M.: Rings of constants for $k$-derivations in $k[x_1,\ldots,x_n]$. J. Math. Kyoto Univ. 28 (1988), 111-118. | DOI | MR

[8] Nowicki, A., Zieliński, J.: Rational constants of monomial derivations. J. Algebra 302 (2006), 387-418. | DOI | MR | Zbl

[9] Ossowski, P., Zieliński, J.: Polynomial algebra of constants of the four variable Lotka-Volterra system. Colloq. Math. 120 (2010), 299-309. | DOI | MR | Zbl

[10] Zieliński, J.: Factorizable derivations and ideals of relations. Commun. Algebra 35 (2007), 983-997. | DOI | MR | Zbl

[11] Zieliński, J.: The five-variable Volterra system. Cent. Eur. J. Math. 9 (2011), 888-896. | DOI | MR | Zbl

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