Liouvillian Integrability Versus Darboux Polynomials

In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y,\, \dot{y}=-f(x)y-g(x)$$\end{document} with degf>degg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg f>\deg g$$\end{document} is not Liouvillian integrable.


Introduction and Statement of the Main Results
One of the classical problems in the dynamical systems is to determine when a system has first integrals. The existence of first integrals together with other invariants like constants of motion, conserved quantities, integrating factors, Jacobian multipliers and Lie symmetries, provide different techniques for studying the dynamics of differential equations coming from different topics such as celestial mechanics, physics, engineering, biology and so on. But in this note we put our attention on the Liouvillian integrability and its related invariants.
Consider complex polynomial differential systemṡ where the dot denotes derivative with respect to the independent variable t which can be real or complex, and P(x) = (P 1 (x), . . . , P n (x)) is an n dimensional vector valued polynomial. For polynomial differential systems (1) there are three related notions: Darboux polynomial, Darboux integrability and Liouvillian integrability. In this note we will present some relations among these three notions.
A function H (x) is a first integral of system (1) if it is continuous and defined in C n except perhaps in a zero Lebesgue measure subset 1 , it is not locally constant on any positive Lebesgue measure subset of C n \ 1 , and H (x) is constant along each orbit of system (1) in C n \ 1 .
A function H (x) is a Darboux first integral of system (1) if it is a first integral and is a function of Darboux type, i.e. it is of the form where r j ∈ C, f j , g, h are polynomials for j = 1, . . . , k, and g and h are coprime. Let (C(x), ) be the differential field given by the field of all rational functions C(x) in the variable x ∈ C n , and = {∂/∂ x 1 , . . . , ∂/∂x n } be a set of derivations on the field C(x). A differential field (L , * ) is a Liouvillian extension of (C(x), ) if there is a tower of fields of the form Note that the derivations * on L restricted to C(x) coincide with the derivations , because this field extension has added only new functions in x but not new variables. The constants of the differential field (L , * ) are all those elements annihilated by all the derivations of * . When the set of constants of a Liouvillian extension (L , * ) of (C(x), ) is C, then one can see by induction that each element of L is an analytic function on a dense open set on C n , see for more details the paper of Singer [36] and the references quoted there. We call such a function a Liouvillian function of n variables.
A function H (x) is a Liouvillian first integral of system (1) if it is a first integral of this system and it is a Liouvillian function.
System (1) is Liouvillian integrable, if it has n − 1 functionally independent Liouvillian first integrals in C n . Recall that k functions H 1 (x), . . . , H k (x) are functionally independent in C n if the matrix k × n with rows the gradients ∇ H 1 , . . . , ∇ H k has rank k for all x ∈ C n except perhaps in a zero Lebesgue measure subset. System (1) is Darboux integrable, if it has n − 1 functionally independent Darboux first integrals in C n .
Here C[x] denotes the ring of all polynomials in the variable where X is the vector field associated to system (1), i.e.
As usually the divergence of the vector field is defined by A C 1 non-zero function J is a Jacobian multiplier of system (1) if it is defined in C n except perhaps in a zero Lebesgue measure subset 1 , and satisfies div(J X ) = 0, i.e. X (J ) = −J divX , x ∈ C n \ 1 .
A Jacobian multiplier is Darboux if it is of Darboux type. As convention, if system (1) is two dimensional, a Jacobian multiplier is called an integrating factor. Darboux theory of integrability was started in 1878 by Darboux [7,8], who provided a method to construct first integral using Darboux polynomials. This theory has been greatly developed in different aspects, see for instance [2,5,6,11,12,15,24,[27][28][29][30]44] and the references therein. The Darboux theory of integrability has also been broadly applied to study dynamics of differential systems, for instance the center problem, limit cycles and global dynamical analysis, see for example [5,10,19,20,33,34,[40][41][42] and the references quoted there. We must mention that during many years after the works of Darboux at the last part of the XIX century and of Lagutinskii at the beginning of the XX century (see [9]), the now called Darboux theory of integrability remained almost forgotten until the paper of Prelle-Singer [32] appeared in 1983. But this paper, published in the Transaction of the AMS, not a specialized journal in differential equations, remained almost unknown to specialists until Dana Schlomiuk's papers [33,35] and her lectures in Montreal, Hasselt and Luminy attracted the attention of the specialists.
Singer [36] in 1992 proved that if a planar polynomial differential system is Liouvillian integrable then it has a Darboux integrating factor, and vice versa, see also [4].
Zhang [43] proved that Liouvillian integrability of higher dimensional polynomial differential systems implies the existence of Darboux Jacobian multipliers. In general Darboux integrability depends on the existence of Darboux polynomials and exponential factors. There are plenty of results studying the existence of Darboux polynomials, see for example [1,16,25,26,34,[37][38][39]45] and the references mentioned there.
Our first result shows that a Liouvillian integrable polynomial differential system under convenient conditions on the degree of the system implies the existence of Darboux polynomials.
Theorem 1 Let k j be the degree of the polynomial P j (x 1 , . . . , x n ) with respect to the variable x k for some k ∈ {1, . . . , n}. If the polynomial differential system (1) has a Darboux Jacobian multiplier and if there exist k, ∈ {1, . . . , n} such that k > max{k 1 , . . . ,k , . . . , k n } + 1, then system (1) has a Darboux polynomial. Herê k denotes the absence of k .
Combining Theorem 1.2 of [43], we can get the next result. (1) is Liouvillian integrable and satisfies the condition C, then system (1) has a Darboux polynomial.

Corollary 2 If the polynomial differential system
We remark that Theorem 1 and Corollary 2 are extension of Theorem 1 and Corollary 2 of [13] from two dimensional polynomial differential systems to any finite dimensional polynomial differential systems.
Our next result shows that Theorem 1 and Corollary 2 in general do not hold if k ≤ max{k 1 , . . . ,k , . . . , k n } + 1 for all k, ∈ {1, . . . , n}. (1)  If systems (1) satisfy the condition C, and have a Darboux polynomial and are integrable, then they may not be Liouvillian integrable as the following proposition shows.

Proposition 4 There exist integrable systems (1) satisfying the condition C, which have a Darboux polynomial and a first integral which is not Liouvillian.
As an application of Theorem 1 and Corollary 2 to Liénard differential system, we get the following result.
Theorem 5 The polynomial Liénard differential systemṡ with f and g polynomials in x, are not Liouvillian integrable provided that deg f > deg g and g(x) ≡ 0.
We remark that under the assumptions of Theorem 5 we can apply the result of Odani [31], and then we know that system (3) has no invariant algebraic curves. But, then Theorem 5 does not follows from Singer result [36] because it can exist an integrating factor formed by exponential factors coming from the multiplicity of infinity.
Recently Llibre and Valls [23] proved that system (3) is not Liouvillian integrable provided that deg g = deg f + 1. We note that Theorem 5 can allow deg g = 0, whereas deg g > 0 in [21,22]. Here we provide a different and easier proof than [21,22] on the non-Liouvillian integrability for the class of Liénard differential systems studied in Theorem 5.
The rest of this paper is devoted to prove the stated results.

Proof of the Main Results
For proving our theorems we need the following result, which is well known for the mathematicians working in the Darboux theory of integrability, but as far as we know there is not published a proof. Proof Recall that X is the vector field associated to system (1). By assumption F is a Jacobian multiplier, so we have X (F) = −FdivX . Writing this equality in components we obtain From this equality we get that for any j ∈ {1, . . . , m} the polynomial f j divides g 2 X ( f j ). This implies that f j divides g or X ( f j ). If the latter happens, there exists , and consequently f j is a Darboux polynomial. If the former happens, we can write g(x) = g 0 (x) f l j (x), where g 0 is a polynomial relative prime with f j , and l is a positive integer. Equating the power of f j in the components of (4) we get that f l j divides hX (g). Since g and h are relatively prime and f j is a factor of g, we must have f l j divides X (g). In addition This shows that f j divides X ( f j ). So we have proved that all f j 's are Darboux polynomials of system (1).
Since each f j is a Darboux polynomial, we can erase the factor f r 1 1 . . . f r m m from equation (4). Then we get from this resulting equation that g divides hX (g). Consequently g divides X (g) because g and h are relatively prime. This shows that g is a Darboux polynomial. This completes the proof of the proposition.
Proof of Theorem 1 Without loss of generality we prove the theorem for = n. Set x = (x 1 , . . . , x n−1 ), and write system (1) in the forṁ where the p k n ≡ 0. By assumption system (1) has a Darboux Jacobian multiplier, denoted by J . Recall that a Darboux Jacobian multiplier is a Jacobian multiplier of the form (2). From Proposition 6, if system (5) has no Darboux polynomials then the Darboux Jacobian multiplier J must be of the form n with r ≥ 0 and h r (x) ≡ 0. By definition of Jacobian multiplier we get that By equating the coefficients of x k n +r −1 n , we get from assumption k n > max{k 1 , . . . , k n−1 } + 1 that (r + k n ) p (n) k n (x)h r (x) = 0. This is a contradiction because r ≥ 0, k n > 0, and p (n) k n (x)h r (x) = 0. This completes the proof of the theorem.
The next result was proved in [43].

Theorem 7 If system (1) is Liouvillian integrable, then it has a Darboux Jacobian multiplier.
Proof of Corollary 2 Its proof follows immediately from Theorems 1 and 7.

Proof of Proposition 3
We first prove statement (a). Consider the following polynomial differential system in R 3 with k, > 1 even. Clearly this system satisfies the degree condition of the proposition. We can check that system (7) has the first integrals Clearly they are functionally independent. Moreover H 1 and H 2 are Liouvillian because they are obtained by taking exponential and integrating from polynomial functions, and each of these steps belongs to the tower elements in the Liouvillian field extension of C(x). Next we prove that system (7) has no Darboux polynomials. We should mention that we cannot directly use the two first integrals H 1 and H 2 to prove our arguments, even through any other first integral is functionally dependent of H 1 and H 2 , because we do not know if system (7) has other first integrals, whose level surfaces contain invariant algebraic surfaces.
Let Y be the vector field associated to system (7). If F(x, y, z) is a Darboux polynomial, then there exists a polynomial K (x, y, z) such that Y(F) = K F. Write F and K as polynomials in the variable x, i.e.
with m, r ≥ 0 integers, and the coefficients polynomials in y and z. Then we get from Without loss of generality we can assume that f m (y, z) ≡ 0. Equating the coefficients of x k in (9) with k > m we get that k 1 (y, z) = · · · = k r (y, z) ≡ 0. Equating the coefficients of x m in (9) gives For the coefficients of x s in (9), s = 0, 1, . . . , m − 1, we have If f m = constant we set f m (y, z) = g 0 (z) + g 1 (z)y + · · · + g q (z)y q , with q a nonnegative integer and the g j 's polynomials in z for j = 0, 1, . . . , q such that g q (z) ≡ 0. Substituting the expression of f m into Eq. (10) and equating the coefficients of y j for j ≥ q, we get that k 0 (y, z) = k 0 (z) and Its general solution is where C is a constant. So this linear equation has a polynomial solution only in the case k 0 (z) = (2m + 3q)z, and g q (z) = C is a constant. Consequently q > 0 because by assumptions f m (y, z) is not a constant. Equating the coefficient of y q−1 in equation (10) we get This linear equation has the general solution where c is the integrating constant. Since is even, we set = 2σ , then by [ where (2σ + 1)!! = (2σ + 1)(2σ − 1) . . . 3 · 1. In order that g q−1 (z) be a polynomial, we must have c = 0 and qC = 0, a contradiction because neither q nor C can be zero. If f m (y, z) = K equal to a non-zero constant, then m > 0, otherwise F(x, y, z) is a constant. We get from (10) that k 0 (z) = 2mz. Now equation (11) Since k > 1 the solution of this last equation cannot be a constant. Set Then we get from Eq. (12) that system (14) has the Darboux polynomial η(x, y, z). This completes the proof of the proposition.
We remark that for polynomial differential systems in R ṅ where k j , j 's are positive integers and no less than 2, using similar arguments to the ones of the proof of Proposition 3 we can prove that systems (17) are Liouvillian integrable, and have or may not have a Darboux polynomial for suitable choices of the values of k j and j . But we need to discuss more cases, so we omit it.
Clearly, this system satisfies the degree condition of the proposition and has the Darboux polynomial F(x, y) = y. It was proved in [3] that system (18) has the non-Liouvillian first integral To obtain such first integral we transform the Abel polynomial differential system (18) into the equation Note that the birational map transforms equation (19) into the Riccati equation dY d X = Y 2 −X . In this case the system of the Abel equations has the non-Liouvillian first integral, where the transcendental functions in the first integral are in the variable x 2 − 1 y . The change has been obtained by taking x 2 − 1 y as a new variable. Proof of Theorem 5 By contrary we assume that system (3) is Liouvillian integrable. Let H (x, y) be a Liouvillian first integral of system (3). Take the change of variables System (3) is transformed tȯ Set G(x 1 , x 2 ) = H (x 1 , x 2 − F(x 1 )). Then G is Liouvillian and it is a first integral of system (21). Indeed, direct calculations show that where ∂ x denotes the partial derivative with respect to x, and in the last equality we have used the fact that H (x, y) is a first integral of system (3).
By assumption deg f > deg g, we have deg F > deg g + 1. So we get from Corollary 2 that system (21) has a Darboux polynomial. Let M(x 1 , x 2 ) be a Darboux polynomial of system (21) with the associated cofactor K (x 1 , x 2 ). Set N (x, y) = M(x, y + F(x)), L(x, y) = K (x, y + F(x)).
Then N and L are both polynomials. We claim that N (x, y) is a Darboux polynomial of system (3) with the associated cofactor L(x, y). Indeed, we get from the change (20) that where in the third equality we have used the fact that M(x 1 , x 2 ) is a Darboux polynomial of system (21) with cofactor K (x 1 , x 2 ). This proves that system (3) has the Darboux polynomial N (x, y).
In addition, Odani [31] proved that if f, g ≡ 0, deg f ≥ deg g and g/ f ≡ constant, then system (3) has no Darboux polynomials. Since our theorem is under the assumption of the Odani's theorem, system (3) cannot have a Darboux polynomial. This contradiction verifies that system (3) is not Liouvillian integrable. This completes the proof of the theorem.