Polynomial algebra of constants of the four variable Lotka–Volterra system

Abstract

We describe the ring of constants of a specific four variable Lotka-Volterra derivation. We investigate the existence of polynomial constants in the other cases of Lotka-Volterra derivations, also in n variables.

Rings of constants of generic 4D Lotka-Volterra systems

“…. = C n = 1 of our paper [9]. As an obvious consequence of the fact that x i | d(x i ), for i = 1, .…”

“…, n, we obtain the following proposition. We formulated Lemma 4.4 in [9] without a proof. Note that there is no assumption on the coefficients C i in this lemma.…”

“…Lotka-Volterra derivations with positive rational coefficients are investigated for instance in [4], [5], [9], [11].…”

Rings and fields of constants of cyclic factorizable derivations

“…The first inspiration for the presented series of articles (some of them are joint works with Hegedűs and Ossowski) was the publication [20] of professor Nowicki and professor Moulin Ollagnier. The fundamental problem investigated in that series of articles concerns rings of polynomial constants ( [26], [28], [33], [29], [8]) and fields of rational constants ( [30], [31], [32]) in various classes of cyclic factorizable derivations. Moreover, we investigate Darboux polynomials of such derivations together with their cofactors ( [33]) and applications of the results obtained for cyclic factorizable derivations to monomial derivations ( [31]).…”

“…[2]). The work [26] presents a description of the ring of constants of the Volterra derivation in four variables, which in this case has three algebraically independent generators:…”