The field of rational constants of the Volterra derivation

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.


INTRODUCTION
The main result of the paper is Theorem 2, which gives the description of the field of rational constants of the four-variable Volterra derivation.The motivations of our study are the following: • applications of Volterra and Lotka-Volterra systems in population biology, laser physics, and plasma physics (see, for instance, [1,3]); • Lagutinsky's procedure of the association of the factorizable derivation (examples of such derivations are Lotka-Volterra derivations) with any given derivation; • link to the invariant theory (for every connected algebraic group G ⊆ GL n (k) there exists a derivation d such that k[X] G = k[X] d , see [7]).
Let us fix some notations: k -a field of characteristic zero, N -the set of nonnegative integers, We call R d = ker d the ring of constants of the derivation and there exists exactly one derivation where the polynomials f i are of degree 1 for i = 1, . . ., n.We may associate the factorizable derivation with any given derivation of k[X], and that construction helps to establish new facts on constants, especially rational constants, of the initial derivation (see, for instance, [6,8]).
There is no general effective procedure for determin- nor even deciding whether it is finitely generated (it may not be finitely generated for n ≥ 4, see [5]).Even for a given derivation the problem may be difficult, see for instance counterexamples to Hilbert's fourteenth problem (all of them are of the form k[X] d ; however, it took more than half a century to find at least one of them, for more details we refer the reader to [5,7]) or Jouanolou derivations (where the rings of constants are trivial, see [6,7]).
We will call a polynomial g ∈ k[X] strict if it is homogeneous and not divisible by the variables x 1 , . . . ,x n .For α = (α 1 , . . ., α n ) ∈ N n , we denote by where X α is a monomial and g is strict.
A nonzero polynomial f is said to be a Darboux polynomial of a derivation δ We will call Λ a cofactor of f .Thus, constants of a derivation δ are precisely its Darboux polynomials with cofactor 0. Denote by k for every m.Since d is a homogeneous derivation of degree 1, the cofactor of each homogeneous polynomial is a linear form.

then its cofactor is a linear form with coefficients in N.
If C i = 1 for all i, then we call d a Volterra derivation.Such derivations were investigated for example in [2], [4], and [10].
Lemma 2. All strict Darboux polynomials of the 4-variable Volterra derivation are its constants.

THE FIELD OF RATIONAL CONSTANTS
We show how to use the results of the previous section to determine the field of rational constants.For any derivation δ : k[X] → k[X] there exists exactly one derivation δ : k(X) → k(X) such that δ|k[X] = δ .By a rational constant of the derivation δ we mean the constant of its corresponding derivation δ : k(X) → k(X).The rational constants of δ form a field.
For simplicity, we write δ instead of δ .
Throughout the rest of this paper we assume n = 4 and that is, d is the fourvariable Volterra derivation.We know the ring of polynomial constants of d (and we use this in the proof of Theorem 2).
A generalization of Theorem 1 can be found in [11].We also need the following facts.
where a, b, c, e ∈ k.By Proposition 2 all homogeneous components of both f and g are also Darboux polynomials of d with the same cofactor Λ.Consider one of these components h = x r 1 x s 2 x t 3 x u 4 h, where h is strict.It is easy to prove, using induction on r + s + t + u, that By Proposition 3, h is a Darboux polynomial of d.Hence, by Lemma 2, h is a constant of d.Therefore we have that is, h is a Darboux polynomial of d with cofactor (s − u)x 1 + (t − r)x 2 + (u − s)x 3 + (r −t)x 4 which, on the other hand, must be Λ.Consequently we have equations: Therefore a, b, c, e ∈ Z. Assume first that a > 0. Then s = u + a > 0, which means that x 2 | h.Since h was chosen as an arbitrary homogeneous component of f or g, we have x 2 | f , x 2 | g, a contradiction with gcd( f , g) = 1.Similarly, assuming a < 0, we get u = s − a > 0 yielding a contradiction x 4 | f , x 4 | g.This proves a = c = 0, and one can use similar arguments for proving b = e = 0. We have proven that Λ = 0, that is, both f and g are constants of d.Thus, by Theorem 1, f , g ∈ Note that in view of Theorems 1 and 2, if d is the four-variable Volterra derivation, then k(X) d is the field of fractions of k[X] d (which is not true in general).

CONCLUSIONS
If δ is a derivation of k(X) such that δ (x i ) = f i for i = 1, . . ., n, then the set k(X) δ \ k coincides with the set of all rational first integrals of a system of ordinary differential equations dx i (t) dt = f i (x 1 (t), . . ., x n (t)), where i = 1, . . ., n (for more details we refer the reader to [7] 1.6).Therefore, we described both all rational constants of the four-variable Volterra derivation and all rational first integrals of its corresponding system of differential equations.We believe that Lemma 1 would be useful for solving the problem also for arbitrary fourvariable Lotka-Volterra derivations.

Theorem 2 .
be a derivation and let f and g be nonzero relatively prime polynomials from k[X].Then δ ( f g ) = 0 if and only if f and g are Darboux polynomials of δ with the same cofactor.Proposition 2. ([7] 2.2.3).Let δ be a homogeneous derivation of k[X] and let f ∈ k[X] be a Darboux polynomial of δ with the cofactor Λ ∈ k[X].Then Λ is homogeneous and each homogeneous component of f is also a Darboux polynomial of δ with the same cofactor Λ.Proposition 3. ([7] 2.2.1).Let δ be a derivation of k[X].If f ∈ k[X]is a Darboux polynomial of δ , then all factors of f are Darboux polynomials of δ .Now we can describe the field of rational constants of d.If d is the four-variable Volterra derivation, then k