Darboux polynomials, balances and Painlevé property

For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years.


INTRODUCTION AND STATEMENT OF THE MAIN RESULTS
The Painlevé property appears in studying the general solutions of the differential equations viewed as functions of complex time. More precisely, when the solutions are single-valued on its maximum domain of analytic continuation, we say that the system has the Painlevé property. In other words, a differential system has the Painlevé property if its general solution has no movable critical singularities, for more details see [3]. This property imposes strong conditions that, despite the fact that it has not been proved, one believes that in this case the system is integrable. However, there is no precise algorithm to decide whether a system has the Painlevé property, and only necessary conditions can be obtained, called the Painlevé test. Most of the systems do not satisfy the Painlevé test, but there is a lot of information concerning the global behavior of the system that we can obtain from the local analysis around the singularities in complex time and the lack of meromorphicity can be used to prove the nonintegrability of the system with meromorphic first integrals.
For more than half a century after its development, the Painlevé theory for differential equations was considered an interesting and important, but perhaps slightly old-fashioned, part of the theory of special functions, and little attention was paid to it until the early 1980s when its relation to soliton theories was discovered. Since then there has been a huge amount of work relating the Painlevé property to different branches of differential systems such as the integrability of PDEs, the rational and polynomial integrability of ODEs etc. However, very little is known about its relation to the Darboux theory of integrability for polynomial differential systems. The main aim of this paper is to focus on the connections between the existence of Darboux polynomials, the Painlevé property and the Kovalevskaya exponents (introduced by Sophia Kowalevskaya to compute the Laurent series solutions of rigid body motion).
In order to state the main results of the paper, we consider a polynomial differential system of the form and P (x) = P 1 (x), . . . , P n (x) and P i ∈ C[x 1 , . . . , x n ] for i = 1, . . . , n. As usual, C denotes the set of complex numbers, and C[x 1 , . . . , x n ] denotes the polynomial ring over C in the variables x 1 , . . . , x n . Here t can be real or complex. The maximum of the degrees of the polynomials P i for i = 1, . . . , n is called the degree of the polynomial differential system (1.1). Assume that there exists a solution of the form where p = (p 1 , . . . , p n ), α = (α 1 , . . . , α n ) = (0, . . . , 0), α i ∈ C and p i ∈ R are given by one of the nonvanishing solutions of the algebraic equation For a given p there may exist different sets of values of α, called balances. The Kowalevskaya matrix associated to a balance α is where, as usual, DP (α) denotes the Jacobian matrix of P evaluated at α and diag(p 1 , . . . , p n ) denotes the matrix whose diagonal is equal to (p 1 , . . . , p n ) and zeroes in the rest. The eigenvalues of the matrix M are called the Kovalevskaya exponents of the balance α and are denoted by ρ = (ρ 1 , . . . , ρ n ). It can be shown that there always exists a Kowalevskaya exponent equal to −1 related to the arbitrariness of the origin of the parameterizations of the solution by the time. The eigenvector associated to the eigenvalue ρ 1 = −1 is pα = (p 1 α 1 , . . . , p n α n ). For more details see [3] or [2]. In Section 2 we recall how to compute the solutions of the form (1.2) when the polynomial differential system is quasi-homogeneous.
where K ∈ C[x 1 , . . . , x n ] is called the cofactor and has degree at most m − 1 if m is the degree of the polynomial differential system (1.1). As usual, ∇F denotes the gradient of the function F . We recall that F is a Darboux polynomial if and only if F (x) = 0 is an invariant hypersurface of system (1.1), i.e., if a solution of system (1.1) has a point on the hypersurface F (x) = 0, then the whole solution is contained in this hypersurface. A polynomial F is said to be weight-homogeneous if there exists d ∈ Q and s = (s 1 , . . . , s n ) ∈ Q n such that for an arbitrary positive real a we have Here d is called the weight degree of the polynomial F and s is the weight exponent of F .
The first result of this paper is the following.

Theorem 1.
Assume that the polynomial differential system (1.1) admits the particular solution , then ∇F (α) = 0 and its cofactor K cannot be constant.
The proof of Theorem 1 is given in Section 4. Theorem 1 is closely related to Theorem 5.4 of [3], which is due to Yoshida (see [7,8]) and states: Under the assumptions of Theorem 1, if I(x) is a weight-homogeneous first integral of weight degree d of system (1.1) satisfying that ∇I(α) = 0, then d is a Kowalevskaya exponent of the matrix M (α) given in (1.4).
For the second result we need some more definitions and notation. We can write the polynomial P i (x) for i = 1, . . . , n in the form i (x) is a weight-homogeneous polynomial of weight exponent p ∈ Q n with weight degree Assume thatẋ = P (0) (x) has the solution x = αt p where t = t − t * for some complex t * , and α ∈ C n with |α| = |α 1 | + · · · + |α n | = 0. Then we say that the polynomial differential system (1.1) admits a dominant balance {α, p}.
We note that any balance of system (1.1) is a dominant balance taking P (0) i = P i . An example of a polynomial differential system with a dominant balance can be found in Section 3. Now we study the relation between the Kovalevskaya exponents and Darboux polynomials.
Assume that the polynomial differential system (1.1) admits a dominant balance {α, p} such that the Kovalevskaya matrix diagonalizes. Then the following statements hold.
Moreover, we explore some connections between the Painlevé property and Darboux polynomials. The first result connecting the Painlevé property with the Darboux polynomials is the following.
Theorem 3. Assume that the polynomial differential system (1.1) satisfies the Painlevé property. Then if the system has a Darboux polynomial, its cofactor K must satisfy K(α) ∈ Z for all balances α of the system.
We also consider a kind of converse result of Theorem 3.

Theorem 4.
Assume that the polynomial differential system (1.1) admits a dominant balance {α, p} and it has a Darboux polynomial with cofactor K such that K (0) (α) ∈ Z. Then system (1.1) cannot satisfy the Painlevé property.
The proofs of Theorems 2, 3 and 4 are given in Section 5. See Section 6 for some examples of systems satisfying the conditions of Theorems 1, 2, 3 and 4 and the concluding section for some comments on our results.

QUASI-HOMOGENEOUS POLYNOMIALS DIFFERENTIAL SYSTEMS
The polynomial differential system (1.1) is quasi-homogeneous if there exists s = (s 1 , . . . , s n ) ∈ N n and r ∈ N such that for an arbitrary positive real a we have P i (a s 1 x 1 , . . . , a sn x n ) = a s i −1+r P i (x 1 , . . . , x n ) for i = 1, . . . , n. We call s = (s 1 , . . . , s n ) the weight exponent of system (1.1) and r the weight degree with respect to the weight exponent s. In particular, in the case that s = (1, . . . , 1) we say that system (1.1) is a homogeneous polynomial differential system of degree r. If a polynomial differential system (1.1) is quasi-homogeneous with weight exponent s and weight degree r > 1, then the system is invariant under the change of variables where This fact implies that in this case there exist solutions of the form (1.2) with p i = s i /(1 − r) for i = 1, . . . , n and the coefficients α i satisfying (1.3). The integrability of quasi-homogeneous polynomial differential systems has been investigated by several authors, see, for instance, [2,3,5,6,[8][9][10].

DOMINANT BALANCES
Consider the polynomial differential system (1.1). We select a suitable weight change of variables denoted by X = (X 1 , . . . , X n ) = a s x = (a s 1 x 1 , . . ., a sn x n ) where s i ∈ N for i = 0, . . . , n such that (1.1) becomesẊ where r m = degree(P ) and P i,m−j is a weight-homogeneous polynomial. In particular, i (x) for j = 1, . . . , r. In this case p = (s 1 , . . . , s n ). Consider the Lorenz systemẋ where s, r, b are real parameters and s = 0. We make the change of variables

PROOF OF THEOREM 1
We first introduce two auxiliary results. Proof. Let HF be the Hessian of the matrix of the polynomial F . To show thatū is a solution of the adjoint equation, we compute its time derivative. Note thaṫ This completes the proof of the lemma.
The following lemma will be the central tool in the nonintegrability results. Its proof uses Lemma 5. Proof. We compute the time derivative of I and show that it is zero. Indeed, by Lemma 5 This completes the proof of the lemma. Now we continue with the proof of the theorem. We proceed by contradiction. Consider system (1.1) which admits the particular solution x = αt p . Assume that it has a weight-homogeneous Darboux polynomial F (x) such that ∇F (α) ≡ 0 and with cofactor K = k 0 . We can apply Lemma 6. The existence of F (x) implies the existence of a first integral I = e −k 0 t ∇F (αt p )u ≡ 0 for the variational equationu = DP (αt p )u.

The general solution of the variational equation is of the form
is a polynomial in log t and can be expressed in terms of the generalized eigenvectors of the Kowalevskaya matrix K = Df (α) − diag(p 1 , . . . , p n ) and ρ i are the Kowalevskaya exponents, for more details see Section 3.8.2 of [3]. This general solution contains n arbitrary parameters. We can therefore evaluate I on this solution to obtain If K is semisimple (i. e., it can be diagonalized), then k = n and the eigenvectors β (i) form a set of n linearly independent vectors of which at least one, say i = j, is such that ∇F (α)β (j) = 0, otherwise I would be zero. Since I is constant in time, we get a contradiction.
If K is not semisimple, then there is a complete set of generalized eigenvectors γ (1) , . . . , γ (n) for K for which at least one of them satisfies ∇F (α)γ (i) = 0 and the contradiction follows. This completes the proof of the theorem.

PROOF OF THE REMAINING RESULTS
In this section we provide the proof of Theorems 2, 3 and 4. To prove Theorem 2, we will use the following theorem, which is Theorem 5.7 in [3]. We continue to decompose any polynomial as in (1.5).
Theorem 7. Assume that the polynomial differential system (1.1) has a Darboux polynomial F with cofactor K and a dominant balance {α, p} with Kovalevskaya exponents ρ = (−1,ρ) with ρ ∈ C n−1 . Then there exists a vector m = (m 2 , . . . , m n ) of positive integers such that Proof (of Theorem 2). The proof of both statements will be done by contradiction.
To prove Theorem 3, we will use again Theorem 7 and the following result proved in [3]. are integers. Now we apply Theorem 7 and from the notation and definitions introduced there we conclude that and the proof is completed.
Proof (of Theorem 4). It follows from Theorem 7 thatρ · m ∈ Z (we have used the notation and definitions introduced there). Therefore, there must exist at least one Kovalevskaya exponent which is not an integer number, and so it follows from Proposition 8 that system (1.1) cannot satisfy the Painlevé property.

Example for Theorem 1
Consider the systemẋ Note that it is a quasi-homogeneous polynomial differential system with weight exponent s = (1, 1, 2, 2) and weight degree r = 2, see Section 2.
System (6.1) has the Darboux polynomials which are both weight-homogeneous with weight degree 2. The cofactor K 1 of F 1 is −2(x 1 − ix 2 ) and the cofactor K 2 of F 2 is 2(x 1 − ix 2 ). None of these cofactors are constant and ∇F j (α) = 0 for j = 1, 2, as stated in Theorem 1. Note that it is not necessary to compute the α corresponding to p = −s because ∇F j = ( * , * , 1, −i).

Example for Theorem 2
Now consider the Lorenz polynomial differential system (3.1) of Section 3. We recall that a dominant balance for system On the other hand, for a nonnegative integer n the Lorenz system with s = −n/2 and b = 2s has the Darboux polynomial x 2 1 − 2sx 3 with cofactor K = K (0) (α) = n. This Lorenz system satisfies the contrapositive of statement (b) of Theorem 2.

Example for Theorem 3
Consider the polynomial differential systeṁ The general solution of this system is with c 1 and c 2 constant. Since this general solution is single-valued on its maximum domain of analytic continuation in C, system (6.2) satisfies the Painlevé property.

CONCLUSIONS
Theorem 1 provides two necessary conditions for the existence of a Darboux polynomial for the polynomial differential system (1.1), one of them related to the balances of the system. Theorem 2 provides two necessary conditions for the existence of a Darboux polynomial of system (1.1), but now one of these conditions is related to the balances of the systemẋ = P (0) (x) instead of the balances of system (1.1).
Theorems 3 and 4 provide necessary conditions for the existence of a Darboux polynomial of system (1.1) using the Painlevé property.
Theorem 7 due to Goriely provides different necessary conditions for the existence of a Darboux polynomial of system (1.1) distinct to the ones given in Theorems 1-4.