Non-existence Criteria for Laurent Polynomial First Integrals

In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial rst integrals for a general nonlinear systems of ordinary dieren tial equations _ x = f(x); x 2 R n with f(0) = 0. We show that if the eigenvalues of the Jacobi matrix of the vector eld f(x) are Z-independent, then the system has no nontrivial Laurent polynomial


Introduction
A system of differential equations ẋ = f (x) (1) is called completely integrable, if it has a sufficiently rich set of first integrals such that its solutions can be expressed by these integrals.According to the famous Liouville-Arnold theorem, a Hamiltonian system with n degrees of freedom is integrable if it possesses n independent integrals of motion in involution.Here, a single-valued function φ(x) is called a first integral of system (1) if it is constant along any solution curve of system (1).If φ(x) is differentiable, then this definition can be written as the condition dφ dx , f (x) = 0.
Obviously, if no such a nontrivial integral exists, then system (1) is called nonintegrable.So finding some simple test for the existence or non-existence of nontrivial first integrals(in given function spaces, such as those of polynomials, rational, or analytic functions) is an important problem in considering integrability and nonintegrability, see Costin [2], Kozlov [6] and Kruskal and Clarkson [7].
The theory of Ziglin [17] has been proved to be one of the most successful approach for proving non-integrability of the n degrees of freedom Hamiltonian systems.This theory has been shown to be useful for many systems, such as the motion of rigid body around a fixed point [17], homogeneous potentials [16], and a reduced Yang-Mills potentials [5], etc.The technique consists of linearizing a system around particular solutions(forming linear manifolds).The linearized equation is then studied.If its monodromy group is too large(i.e., the branches of solutions have too many values) then the linearized equation has no meromorphic first integrals.It is then shown that this implies that the original system has no meromorphic first integrals.
One of the pioneering authors in considering the non-integrability problems for non-Hamiltonian systems is Yoshida.Using a singularity analysis type method, he was able to derive necessary conditions for algebraic integrability for similarityinvariant system [15].In [3], Furta suggested a simple and easily verifiable criterion of non-existence of nontrivial analytic integrals for general analytic autonomous systems.He proved that if the eigenvalues of the Jacobi matrix of the vector field f (x) at some fixed point are N-independent, then the system ẋ = f (x) has no nontrivial integral analytic in a neighborhood of this fixed point.Based on this key criterion, he also made some further study on non-integrability of general semiquasihomogeneous systems.Some related results can be found in [4,9,10,11,12,13].
However, there are still many systems encountered in physics which do not fall in the set of completely integrable or completely non-integrable systems.Indeed, if a system admits a certain number of first integrals less than the number required for the complete integration, then non-integrability cannot be proved in general.Such systems will be called partially integrable [4].Some works have been done in this direction, see [1,8,14].
The function space we are interested in this paper is the Laurent polynomial ring where P k 1 •••kn ∈ C and A, the support of P (x), is a finite subset of integer group Z n .We will give a simple criterion for non-existence of Laurent polynomial first integrals for general nonlinear analytic systems.The outline of this paper is as follows.We will first give a simple criterion of non-existence of Laurent polynomial first integral for general analytic systems of differential equations in Section 2. Then in Section 3, we consider the partial integrability for general nonlinear systems.Some examples are presented in Section 4 to illustrate our results.

A criterion for non-integrability
Consider an analytic system of differential equations in a neighbourhood of the trivial stationary solution x = 0. Let A denote the Jacobi matrix of the vector field f (x) at x = 0. System (2) can be rewritten as near some neighborhood of the origin x = 0, where f e., they do not satisfy any resonant equality of the following type then system (3) does not have any nontrivial Laurent polynomial integral.
Proof.Since after a nonsingular linear transformation, A can be changed to a Jordan canonical form, for simplicity, we can assume A is a Jordan canonical form, i.e., Suppose system (3) has a Laurent polynomial integral where A is the support of P (x), then P (x) has to satisfy the following partial differential equation where •, • denotes the standard scalar product in C n .Note that P (x) can be rewritten as where P k (x) are homogeneous Laurent polynomials of degree k in x and P l (x) ≡ 0, P p (x) ≡ 0. Substitute ( 6) into ( 5) and equate all the terms in (5) of the same order with respect to x with zero and consider the first nonzero term in (6), we get This means that P l (x) is an integral of the linear system ẋ = Ax.
Make the following transformation of variables where Under the transformation, system (7) can be rewritten as where So Q(y, ) = P l (Cy) is an integral of linear system (8), i.e., dQ dy ( , y), (B + B)y ≡ 0. Since where L, M ∈ Z are certain integers, L ≤ M , Q i (v) are homogeneous form of degree l and Q L (y) ≡ 0, Q M (y) ≡ 0. By ( 9) and ( 10), Q L (y) has to satisfy the following equation Since Thus, a resonant condition of (4) type has to be fulfilled for some nonzero coefficient , which contradicts the conditions of Theorem 2.1.The proof is now complete.

A criterion for partial integrability
By the proof of Theorem 2.1, we can see that if system (3) has a nontrivial Laurent polynomial integral P (x), then linear system (7) must have a nontrivial homogeneous Laurent polynomial integral P l (x).So in general at least one resonant relationship of type (4) must be satisfied, and the set is a nonempty subgroup of Z n .
Proof: Under the transformation x = εy, system (3) can be rewritten as We can also rewrite the integrals P i (x) and Q(x) of system (3) as follows where P i l i +j (x) and Q q+j are homogeneous Laurent polynomials.Therefore P 1 (y, ε), • • •, P s (y, ε) and Q(y, ε) are integrals of system (12).Since under the transformation x = εy, we have 16) By ( 15) and ( 16) we obtain Let H (0) = H.Then the function is an integral of system (12), since Q(y, ε) and P 1 (y, ε), • • •, P s (y, ε) are all integrals of system (12).By ( 13), ( 14), ( 15) and ( 17), it is not difficult to see that the function Q(1) (y, ε) is at least of q + 1 order with respect to ε and can be rewritten as where q 1 ≥ q + 1 is an integer, Q q 1 +j (y) is a homogeneous form of degree q 1 + j.Now Q (1)  q 1 (y) is also an integral of linear system (7).According to the assumptions of the lemma, Q (1)  q 1 = H (1) is also an integral of system ( 12) which is at least of q 1 + 1 degree with respect to ε.
By repeating infinitely this process, we obtain that which is equivalent to the fact that for a certain smooth function F .Lemma 2. Assume A is diagonalizable and rank G = s.If linear system (7) has s homogeneous Laurent polynomial integrals P 1 l 1 (x), • • • , P s ls (x) which are functionally independent, then any other nontrivial homogeneous Laurent polynomial integrals Q q (x) of system (7) is a function of P 1 l 1 (x), • • • , P s ls (x).Proof.For simplicity, we assume A has already a diagonal form diag(λ n are s rational integrals of linear system (7).Furthermore, ω 1 (x), • • • , ω s (x) are functionally independent.
In fact, since is full-ranked.Therefore, it must have a subdeterminant of degree s which is nonzero, without loss of generality, we can assume
The following theorem is the direct result of Lemma 1 and Lemma 2. Theorem 3.1.Assume system (3) has s(s < n) nontrivial Laurent polynomial integrals P 1 (x), • • •, P s (x) and matrix A is diagonalizable.If P 1 l 1 (x), • • • , P s ls (x) are functionally independent and rank G = s, then any other nontrivial Laurent polynomial integral Q(x) of system (3) must be a function of P 1 (x), • • • , P s (x).
where α j , β j and a jk are real constants, β n = 0. Some ecological systems, such as Lotka-Volterra systems, can be reduced to such forms by a linear transformation near an equilibrium.