Darboux theory of integrability for polynomial vector fields in R n taking into account the multiplicity at infinity

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1 Bull. Sci. math. 33 ( Darboux theory of integrability for polynomial vector fields in R n taking into account the multiplicity at infinity Jaume Llibre a,, Xiang Zhang b a Departament de Matemàtiques, Universitat Autònoma de Barcelona, 0893 Bellaterra, Barcelona, Catalonia, Spain b Department of Mathematics, Shanghai Jiaotong University, Shanghai , PR China Received 2 May 2009 Available online 26 June 2009 Abstract Darboux theory of integrability was established by Darboux in 878, which provided a relation between the existence of first integrals and invariant algebraic hypersurfaces of vector fields in R n or C n with n 2. Jouanolou 979 improved this theory to obtain rational first integrals via invariant algebraic surfaces using sophisticated tools of algebraic geometry. Recently in [J. Llibre, X. Zhang, Darboux theory of integrability in C n taking into account the multiplicity, J. Differential Equations, in press] this theory was improved taking into account not only the invariant algebraic hypersurfaces but also their multiplicity. In this paper we will show that if the hyperplane at infinity for a polynomial vector field in R n has multiplicity larger than, we can improve again the Darboux theory of integrability. We also show some difficulties for obtaining an extension of this result to polynomial vector fields in C n Elsevier Masson SAS. All rights reserved. MSC: 34A34; 34C05; 34C4 Keywords: Polynomial vector field in projective space; Darboux integrability; Invariant algebraic hypersurface; Multiplicity * Corresponding author. addresses: jllibre@mat.uab.cat (J. Llibre, xzhang@sjtu.edu.cn (X. Zhang /$ see front matter 2009 Elsevier Masson SAS. All rights reserved. doi:0.06/j.bulsci

2 766 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( Introduction and statement of results In the study of systems of differential equations in R n, if they have a first integral, then their analysis can be reduced in one dimension. But the search for a first integral of a given differential system in R n is generally very difficult. Darboux theory of integrability, established by Darboux [2,3] in 878 and improved by Jouanolou [6] in 979, and Llibre and Zhang [], is a nice theory to find first integrals of a differential system having sufficiently many invariant algebraic hypersurfaces taking into account their multiplicity. This theory has been successfully applied to the study of some physical models (see for instance, [8,9,6,7], and to centers, limit cycles and bifurcation problems of planar systems (see for instance, [5,7,4]. In this paper we find that if the hyperplane at infinity has multiplicity larger than, then we can go further improving the Darboux theory of integrability taking into account the multiplicity of the hyperplane at infinity. We consider in this paper polynomial vector fields in R n.letr[x] be the ring of polynomials in the variable x R n with coefficients in R. Consider the following polynomial vector fields n X = P i (x, x R n, ( x i i= where P i (x R[x] and (P,...,P n =, i.e. they have no common factors. We call d = max{deg P i (x; i =,...,n} the degree of the vector field X. Let C[x] be the ring of polynomials in x R n with coefficients in C. Forf = f(x C[x], we say that {f = 0} R n is an invariant algebraic hypersurface of the vector field X if there exists a polynomial L f C[x] such that X (f = n i= P i f x i = fl f. We call L f the cofactor of f = 0. It is necessary that L f has degree at most d. Assume that f,g C[x] and (f, g =, i.e. relatively coprime. We say that exp(g/f with deg g deg f or deg f = 0isanexponential factor of the vector field X if there exists an L e C[x] of degree at most d such that X ( exp(g/f = exp(g/f L e. We call L e the cofactor of the exponential factor. It is easy to verify that if exp(g/f is an exponential factor and deg f 0, then f = 0 is an invariant algebraic hypersurface. Let C m [x] be the C-vector space formed by polynomials in C[x] of degree at most m. Note that this vector space has dimension σ = ( n+m n. Choose a base v,...,v σ of C m [x], and denote by M σ the σ σ matrix v v 2 v σ X (v X (v 2 X (v σ......, (2 X σ (v X σ (v 2 X σ (v σ where X k+ (v i = X (X k (v i. If det M σ 0 the hypersurface {det M σ = 0} R n is called the m-th extactic hypersurface of X, and the det M σ is called m-th extactic polynomial. Observe that the extactic hypersurface modulus a constant is independent of the choice of the base of C m [x].

3 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( So if f = 0 is an irreducible invariant algebraic hypersurface of degree m of X, then f divides det M σ. This follows from the fact that we can choose f as a member of a base of C m [x] and f divides the whole column in which f is located. Recall that an algebraic hypersurface f = 0 is irreducible if f is an irreducible polynomial in C[x]. We say that an irreducible invariant algebraic hypersurface f = 0ofdegreem has defined algebraic multiplicity k or simply algebraic multiplicity k if det M σ 0 and k is the maximum positive integer such that f k divides det M σ ; and it has no defined algebraic multiplicity if det M σ 0. The matrix (2 already appears in the work of Lagutinskii (see also Dobrovol skii et al. [4]. For a modern definition of the m-th extactic hypersurface and a clear geometric explanation of its meaning, the readers can look at Pereira [2]. In order to use the infinity of R n as an additional invariant hyperplane for studying the integrability of the vector field X, we need the Poincaré compactification for the vector field X, which we summarize in Appendix A. In the chart U (see (4, using the change of variables x = z, x 2 = y 2 z,..., x n = y n z, (3 the vector field X is transformed to X = zp (y z + ( P 2 (y y 2 P (y + + ( P n (y y n P (y, y 2 y n where P i = z d P i (/z, y 2 /z,...,y n /z for i =,...,n and y = (z, y 2,...,y n. We note that z = 0 is an invariant hyperplane of the vector field X and that the infinity of R n corresponds to z = 0 of the vector field X. So we can define the algebraic multiplicity of z = 0 for the vector field X. We say that the infinity of X has defined algebraic multiplicity k or simply algebraic multiplicity k if z = 0 has defined algebraic multiplicity k for the vector field X ; and that it has no defined algebraic multiplicity if z = 0 has no defined algebraic multiplicity for X. In [5] the authors gave a definition on the algebraic multiplicity of the line at infinity for a planar vector field using a limit inside the definition. In fact, the two definitions are equivalent. But ours is easier to be applied to compute the algebraic multiplicity of the line at infinity for a given planar vector field. Let D be an open subset having full Lebesgue measure in R n. A non-constant analytic function H : D R is a first integral of the polynomial vector field X on D if it is constant on all orbits x(t of X contained in D;i.e.H(x(t= constant for all values of t for which the solution x(t is defined and contained in D. Clearly H is a first integral of X on D if and only if X H = 0 on D.Arational first integral of X is a first integral given by a rational function. A Darboux first integral is a first integral of the form ( r exp(g/h, i= f ν i i where f i,g,h C[x] with deg g deg h or deg h = 0, and the ν i s are complex numbers. We remark that for a real polynomial vector field its invariant algebraic hypersurfaces and the exponential factors can be complex, but the Darboux first integral may be real. The following is the main result of this paper, which improves the Darboux theory of integrability in R n taking into account the algebraic multiplicity of the hyperplane at infinity. Theorem. Assume that the polynomial vector field X in R n of degree d>0 has irreducible invariant algebraic hypersurfaces f i = 0 for i =,...,pand the invariant hyperplane at infinity.

4 768 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( (i If some of these irreducible invariant algebraic hypersurfaces or the invariant hyperplane at infinity has no defined algebraic multiplicity, then the vector field X has a rational first integral. (ii Suppose that all the irreducible invariant algebraic hypersurfaces f i = 0 have defined algebraic multiplicity q i for i =,...,pand that the invariant hyperplane at infinity has defined algebraic multiplicity k. If the vector field restricted to each invariant hypersurface including the hyperplane at infinity having algebraic multiplicity larger than has no rational first integral, then the following hold. (a If p i= q i + k N + 2, then the vector field X has a real Darboux first integral, where N = ( n+d n. (b If p i= q i + k N + n +, then the vector field X has a real rational first integral. We note that if the hyperplane at infinity is not taken into account, then Theorem is exactly Theorem 3 of []. Also if the hyperplane at infinity has algebraic multiplicity, then it does not contribute to integrability by comparing Theorem with Theorem 3 of []. In [] we had showed by examples that the assumption on the non-existence of rational first integral of X restricted to an invariant algebraic hypersurface with multiplicity larger than is necessary for the vector field in R n with n>2. In Section 2 we show by an example that if n>2, the additional assumption is also necessary for the infinity having multiplicity larger than. If X is a planar vector field, then this additional assumption about the rational first integral is not necessary. We have the following Corollary 2. Assume that the polynomial vector field X in R 2 of degree d>0 has irreducible invariant algebraic curves f i = 0 with defined algebraic multiplicity q i for i =,...,p and that the invariant straight line at infinity has defined algebraic multiplicity k. Then the following hold. (a If p i= q i + k ( d , then the vector field X has a Darboux first integral. (b If p i= q i + k ( d , then the vector field X has a rational first integral. This paper is organized as follows. In Section 2 we will provide some basic results which will be used in the proof of Theorem. The proof of Theorem is given in Section 3. This proof is similar to the proof of Theorem 3 of [], we provide it here for completeness. In Section 4 we present some examples showing how use the algebraic multiplicity of the hyperplane at infinity for finding first integrals. In Section 5 we explain the difficulties for extending to C n the Darboux theory of integrability taking into account the multiplicity at infinity. Appendix A summarizes the Poincaré compactification of a vector field in R n. 2. Basic results For characterizing the algebraic multiplicity of an invariant algebraic hypersurface f using the exponential factors associated with f, we obtained in [,] the following result. Theorem 3. Assume that the vector field X has an irreducible invariant algebraic hypersurface f = 0 of degree m and that X restricted to f = 0 has no rational first integral. Then f = 0 has algebraic multiplicity k if and only if the vector field X has k exponential factors exp(g i /f i, i =,...,k, where g i C[x] has degree at most im.

5 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( Similar to Theorem 3 we have the following result characterizing the existence of exponential factors associated with the hyperplane at infinity for the compactified vector field p(x of X. Theorem 4. Let X be the expression of the compactified vector field p(x in U. Assume that X restricted to z = 0 has no rational first integral. Then z = 0 has algebraic multiplicity k for X if and only if X has k exponential factors exp(g i /z i, i =,...,k, with g i C i [y] having no factor z. For proving Theorem 4 we need the following result, which provides a relation between the exponential factors of X and those of X associated with z = 0. Lemma 5. For the exponential factors associated with the hyperplane at infinity the following statements hold. (a If E = exp(g(x with g a polynomial of degree k is an exponential factor of X with cofactor L E (x, then E = exp( g with g = z k g( z k z, y 2 z,..., y n z is an exponential factor of X with cofactor L E = z d L E ( z, y 2 z,..., y n z. (b Conversely if F = exp( h(y z k with h R k [y] is an exponential factor of X with cofactor L F, then F = exp(h(x with h(x = x k h( x, x 2 x,..., x n x is an exponential factor of X with cofactor L F = x d L F ( x, x 2 x,..., x n x. Proof. (a Some computations show that ( ( ( g X z k = X g z, y 2 z,...,y n z ( = zp g z 2 y 2 g x z 2 y n g x 2 z 2 x n ( + (P 2 y 2 P g + +(P n y n P g x 2 x n = ( g g P + +P n z x x n = L E. ( = z d X (g = z d L E z, y 2 z,...,y n z This implies that E is an exponential factor of X. (b First we claim that if E = exp(g/z r with g C r [y] is an exponential factor of X with cofactor L E, then its degree is at most d. Now we prove the claim. From the definition of exponential factor we have X (g/z r = L E. This equation can be written as P 2 g y 2 + +P n g y n P ( z g z + y g g 2 + +y n y 2 y n z + rgp = z r L E. If deg g<rthen clearly we have deg L E <d.ifdegg = r then using the Euler s formula for the homogeneous part of degree r of g we get also deg L E <d. This proves the claim.

6 770 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( It follows from the claim that L F is a polynomial of degree d. Direct calculations yield (( X (h = P h x 2 z x 2 h x 2 x n h y 2 x 2 x k y + khxk n ( h h + P 2 + +P n x k y 2 y n = z d ( zp h z y 2P h y 2 y n P h + z d ( P 2 h y 2 + +P n h y n z (k = z d( X (h + kp h z (k = z (d X + kp h y n ( h z k z (k = z (d L F (z, y 2,...,y n = L F. So exp(h is an exponential factor of X. This completes the proof of the lemma. Proof of Theorem 4. It follows from Theorem 3 and Lemma 5. Using a similar proof to that of statement (b of Lemma 5 we can get easily the following result. Proposition 6. If H(y is a rational first integral of X, then H(x = H( x, x 2 x,..., x n x is a rational first integral of X. The following proposition shows that in dimension larger than 2 the assumption in Theorem 4 on the non-existence of rational first integral of X restricted to z = 0 is necessary. Proposition 7. The system ẇ = ( cw x bw 2 w, ẋ = ( + cx y bwxw x 2, ẏ = ( x + (a + cy bwy w xy (4 has the invariant plane w = 0 of multiplicity 4, where a,b,c are real constants. The system restricted to w = 0 has the rational first integral H = y/x. For this system Theorem 4 does not hold. Proof. The argument on the multiplicity can be proved by computing the -st extactic polynomial. The existence of the first integral H = y/x is an easy calculation. We remark that system (4 is the local expression in the chart U 3 of the Poincaré compactified vector field of the Rössler system [3] ẋ = (y + z, ẏ = x + ay, ż = b cz + xz. (5 In Theorem of [8] we proved that system (5 has only either the exponential factor F = exp(u x + u 2 y with u and u 2 not zero simultaneously if a 0, or the exponential factors F and F 2 = exp(z + x 2 /2 + y 2 /2 if a = 0.

7 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( From Lemma 5 we get that system (4 has only the exponential factors associated with w = 0 either E = exp((u x + u 2 y/w with u and u 2 not zero simultaneously if a 0, or E and E 2 = exp((w + x 2 /2 + y 2 /2/w 2 if a = 0. This means that system (4 has no exponential factors of the form exp(g/w 3 with g a polynomial of degree at most 3 and coprime with w. Hence the conclusion of Theorem 4 does not hold. 3. Proof of Theorem (i We can consider the real vector field X as a vector field in C n. Then it follows from the second part of Theorem 3 of Pereira [2] (see also Theorem 5.3 of [] for dimension 2 that if some of the invariant algebraic hypersurfaces has no defined algebraic multiplicity, then the vector field X will have a rational first integral. This first integral may be complex, but its real part and also imaginary part will be real rational first integrals. Similarly if the invariant hyperplane z = 0ofX, i.e. the invariant hyperplane at infinity of X, has no defined algebraic multiplicity, then X has a real rational first integral. Hence from Proposition 6 the vector field X has a real rational first integral. This proves statement (i. (ii Let f i = 0fori =,...,p be the irreducible invariant hypersurfaces having algebraic multiplicity q i. It follows from Theorem 3 that for each f i we have q i exponential factors exp(g ij /f j i, j =,...,q i, where deg g ij j deg f i. From Theorem 4 and Lemma 5 the vector field X has exactly k exponential factors exp(g i (x associated with the hyperplane at infinity with g i polynomials of degree i for i =,...,k. For r =, 2,...,κ := p i= q i + k, denote by k r the κ cofactors of the p invariant algebraic hypersurfaces and of the κ p exponential factors. Recall that each cofactor is a polynomial of degree at most d, where d is the degree of the vector field X. To simplify the notations we denote by F r, with r {,...,κ}, the irreducible invariant algebraic hypersurface or the exponential factor of the vector field X having cofactor k r. Then we have X (log F r = k r for r =,...,κ. (a Since C d [x] is a C-vector space of dimension N and κ N + by the assumption, the κ cofactors k r (x are linearly dependent. So there exist σ,...,σ κ C not all zero such that κr= σ r k r = 0. This shows that X (log(f σ...fσ κ κ = 0. Hence H = log(f σ...fσ κ κ is a first integral of X. Since the vector field X is real, the conjugate H of H is also a first integral of X. Hence H + H is a real first integral of X. On the other hand, H + H = log((f σ F σ...(fσ κ κ F σ κ κ is of Darboux type. This proves statement (a. (b Let τ be the dimension of the vector subspace V generated by {k (x,..., k N+n (x}. Since all k r has degree at most d wehaveτ N. In order to simplify the notations we suppose that τ = N and that k (x,...,k N (x are linearly independent in V. In the case τ<nthe proof would follow using the same arguments. From the proof of statement (a we get that for each s {,...,n} there exists a vector (σ s,...,σ sn C N such that X ( log ( F σ s...f σ sn N F N+s = 0. Thus we get n holomorphic first integrals H s := log ( F σ s...f σ sn N F N+s, (6

8 772 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( for s =,...,n of the vector field X, defined on a convenient full Lebesgue measurable subset Ω of R n. We define r(x = rank{ H (x,..., H n (x} and m = max{r(x: x Ω }. Then there exists an open subset Θ of Ω such that m = r(x for each x Θ. Since the n first integrals H i s must be functionally dependent on any positive Lebesgue measurable subset of Ω,wehavem<n. Without loss of generality we can assume that { H (x,..., H m (x} has the rank m for all x Θ. Therefore, since { H (x,..., H m (x, H k (x} with k {m +,...,n} are linearly dependent for each given x Θ, there exist C k (x,...,c km (x such that H k (x = C k (x H (x + +C km (x H m (x, (7 for k = m +,...,n. By the construction of H i it is easy to check that each component of the vector function H i (x is a rational function. Since H (x,..., H m (x are linearly independent on Θ, by using the Crammer s rule to solve the linear equation (7 we get that C kj are rational functions for j =,...,m. From Theorem 2 of [0] it follows that C kj (x (if not a constant for j =,...,m are first integrals of X. Finally we shall prove that there is some C kj which is not a constant. Assume that all functions C k,...,c km are constants, then we get from (7 that H k (x = C k H (x + + C km H m (x + log C k, where C k is a constant. Hence, for k {m +,...,n} and from (6 we have F σ k...f σ kn N F N+k = C k (F σ...f σ N N F N+ C k...(f σ m...f σ mn N F N+m C km. Thisisin contradiction with the fact that F N+k only appears on the left-hand side of the above equality. So there exist some j 0 {,...,m} and k 0 {m +,...,n} such that C k0 j 0 (x is not a constant. The above proof shows that the vector field X has the rational first integral C k0 j 0 (x. Itmay be complex. Since X is a real vector field, Re C k0 j 0 and Im C k0 j 0 are both rational first integrals of X. This completes the proof of Theorem. 4. Examples In this section we provide some examples showing how to use the algebraic multiplicity of the hyperplane at infinity for proving the existence of a Darboux first integral of a polynomial vector field and for computing this first integral. The first two examples were studied in [5]. Example. Consider the vector field X = / x x 2 / y. It is easy to see that X 2 has the polynomial first integral y + x 3 /3. But we now use this easy example to test our theory. The local expression in U of the compactified vector field p(x is X 2 = z 3 / z ( + yz 2 / y. Choose,y,z as a base of R [y,z], we get the -st extactic polynomial z 5 (4 + 6yz 2. This means that the line at infinity of X 2 has algebraic multiplicity 5. By Theorem the vector field X has a first integral. Some computations verify that exp(x, exp(x 2,exp( 3 x3 + x + y and exp( 4 x4 + xy are four exponential factors associated with the line at infinity. Their corresponding cofactors are k =, k 2 = 2x, k 3 = and k 4 = y. From the proof of statement (a of Theorem, by using the cofactors k and k 3 we get the first integral y + x 3 /3. This example shows that using only the algebraic multiplicity of the line at infinity we can prove the existence of a first integral of X and furthermore compute it.

9 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( Example 2. For the vector field X 2 = x / x + (y x 2 / y, the local expression in U of its compactified vector field p(x 2 is X 2 = z 2 / z / y. Choose the same base as in Example, then the -st extactic polynomial is 2z 3. So the line at infinity of X 2 has algebraic multiplicity 3. This shows that the vector field X 2 has two exponential factors associated with the line at infinity. It is easy to check that exp(x and exp( 2 x2 + y are these two exponential factors with cofactors k = x and k 2 = y, respectively. Obviously the line x = 0 is invariant by the flow of X 2 with cofactor k 3 = and has algebraic multiplicity 3. Some calculations show that exp(y/x and exp(y 2 /x 2 are the related exponential factors with cofactors k 4 = x and k 5 = 2y, respectively. The above computations show that the sum of the multiplicities of the straight lines x = 0 in R 2 and z = 0 at infinity is 6. By Theorem the vector field X 2 has a rational first integral. In fact using the exponential factors exp(x and exp(y/x we get the rational first integral x + y/x. Example 3. Consider the real 3-dimensional polynomial vector field X 3 = + y( 2x + 3z x y + z2 z. Some calculations show that the algebraic multiplicities of the invariant planes y = 0 and z = 0 are and 3, respectively. Moreover it is easy to check that the vector field X 3 restricted to z = 0 has no rational first integral. The local expression in U of the compactified vector field p(x 3 of X 3 is X 3 = w 3 w y( 2 w 3z + w 2 y + z( z w 2 z. The invariant plane w = 0ofX 3 corresponds to the infinity of X 3, and it has algebraic multiplicity 3. Again it is easy to verify that the vector field X 3 restricted to w = 0 has no rational first integral. The total number of the three invariant planes (including the one at infinity taking into account their algebraic multiplicities is 7. Since = 7 > ( = 6, it follows from Theorem that the vector field X 3 has a Darboux first integral. We now compute the first integrals. By some calculations we get that the vector field X 3 has two exponential factors associated with z = 0: e /z with cofactor L = and e (+2xz/z2 with cofactor L 2 = 2x, and two exponential factors associated with the infinity: e x with cofactor L 3 = and e x2 with cofactor L 4 = 2x. Obviously the cofactors corresponding to the invariant planes z = 0 and y = 0areL 5 = z and L 6 = 2x + 3z, respectively. Using the cofactors L and L 3 we get the rational first integral H = x + /z. Combining the cofactors L 3,...,L 6 we obtain the Darboux first integral H 2 = e x2 x yz 3. It is easy to check that the two first integrals are functional independent outside the plane z = 0, where the two first integrals are not defined. So the vector field X 3 is completely integrable. 5. Difficulties on the extension of the theory to C n In this section we shall show that some difficulties appear in extending the Darboux theory of integrability taking into account the multiplicity of the hyperplane at the infinity from R n to C n. A polynomial vector field in C does not have the Poincaré compactification as that in R n,see Appendix A. For studying the multiplicity of the hyperplane at infinity in C n a natural way is to extend the vector field from C n to a vector field of the projective space CP n.

10 774 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( For the planar polynomial vector field in C 2 X = p (x, y x + p 2(x, y y, of degree d, it follows from [6] that its projective vector field in CP 2 is X = P (X,Y,Z X + P 2(X,Y,Z Y, where [X : Y : Z] are the homogeneous coordinates of CP 2 and P i (Z,Y,Z= Z d p i (X/Z, Y/Z for i =, 2. We note that the invariant plane Z = 0ofX corresponds to the infinity of C 2. Denote by C m [X, Y, Z] the set of homogeneous polynomials in the variables X, Y, Z with coefficients in C of degree m. Similar to the case in C n we can choose X, Y, Z as a base of C [X, Y, Z] to define the -st extactic polynomial, and hence to define the multiplicity of Z = 0forX. Two different difficulties will appear trying to extend the previous results on the infinity from vector fields defined in R n to vector fields in C n. The first is that while the restriction of a vector field in R n on Z = 0 has no rational first integral, but the same vector field in C n restricted to Z = 0inCP n may have a rational first integral. For instance the projective vector field of the vector field X 2 in Example 2 of Section 4 is X 2 = XZ X + ( YZ X 2 Y. Clearly Z = 0 is an invariant plane of X 2 in CP 2.OnZ = 0 the projective vector field X 2 has the first integral H = X. So even if we have a similar result to Theorem 4 for using the exponential factors in order to characterize the multiplicity of the invariant plane Z = 0, it also does not work on X 2. On the other hand, we have seen that for the real vector field X 2 the Poincaré compactification works well on this example. In [] we improve the Darboux theory of integrability taking into account the algebraic multiplicity of the invariant algebraic hypersurfaces of C n through the associated exponential factors. We shall see that this is not possible for studying the infinity of vector fields in C n.this will be the second difficulty. Assume that f = 0 is an invariant algebraic hypersurface of degree m with cofactor L f for a vector field X in C n, and that X restricted to f = 0 has no rational first integrals. If f = 0 has algebraic multiplicity r, i.e. r is the maximum number such that f r divides the determinant of the matrix f v 2 v R X (f X (v 2 X (v R M R =......, (8 X R (f X R (v 2 X R (v R where X k+ (v i = X (X k (v i, R is the dimension of the linear space C m [x] with x C n, and {f,v 2,...,v R } is a base of C m [x]. Assume that f = 0 is an invariant algebraic hypersurface with multiplicity r>. Since X r (f = f(x + L f r, where by definition (X + L f r = (X + L f (X + L f r with L f acting as a constant operator and non-commutative with X, we have det M R = f det M R, where M R is different from M R only in the first column with i-th component given by (X + L f i. Since f divides det M R because r>, we have det M R = 0onf = 0. Then the columns of M R,

11 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( denoted by V,...,V R, are linearly dependent on f = 0. So there exist functions α,...,α R not all zero such that for l = 0,,...,R A l := α (X + L f l + R α j X l (v j = 0 onf = 0, (9 j=2 where X 0 = identity. Assume that α 0. Then we can take α =, and we have A l := (X + L f l + q α ij X l (v ij = 0 onf = 0, l= 0,,...,R, (0 j= with at least one of the α ij not zero, where V i...,v iq are linearly independent. Set some α ir 0 with r {,...,q}. Then (X + L f A l A l+ = q ( X (αij + L f α ij X l (v ij = 0 j= on f = 0, l= 0,,...,R 2. It follows that X (α ij + L f α ij = 0onf = 0. So we have X (α ij /α ir = 0onf = 0. This implies that there exist constants c ij for j r such that α ij = c ij α ir on f = 0forj {,...,r, r +,...,q}. Hence from the first two equations of (0 we have R R + α ir c s v s 0 and L f + α ir c s X (v s 0 onf = 0, s=2 s=2 where c ir = and c s = 0fors/ {i,...,i q }. Set g = R s=2 c s v s.wehaveg 0 and X (g g L f 0onf = 0. By the Hilbert s Nullstenesatz and the fact that f is irreducible there exists a polynomial L of degree necessarily no more than d such that X (g = g L f + fl, where d is the degree of the vector field X. This proves that exp(g /f is an exponential factor of X with the cofactor L. Now we shall see that here taking into account the infinity in C n we cannot control the multiplicity of Z = 0 through the associated exponential factors. Due to these difficulties we have studied in this paper only the extension of the Darboux theory of integrability taking into account the multiplicity of the hyperplane at infinity for vector fields of R n. In the projective space CP 2 we try to establish a similar result to Theorem 4, but in its proof we need to use the linear dependence of the columns of the matrix restricted to Z = 0inthe definition of the -st extactic curves. For instance, if Z = 0 is an invariant plane of multiplicity at least 2 for the projective vector field X, then Z divides the determinant of the matrix X Y M = 0 X (X X (Y. 0 X 2 (X X 2 (Y This means that the columns of the matrix M are linearly dependent on Z = 0. Denote by V,V 2,V 3 the three columns of the matrix M. Assume that there exist well-defined functions α,α 2,α 3 not all zero on Z = 0inCP 2 such that A l = α X l ( + α 2 X l (X + α 3 X l (Y = 0 onz = 0forl = 0,, 2. (

12 776 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( If α 0 and the columns V 2 and V 3 are linearly independent on Z = 0, we assume without loss of generality that α =. Then we have X (A l A l+ = X (α 2 X l (X + X (α 3 X l (Y = 0 onz = 0, l= 0,. (2 We claim that (X, X (X τ and (Y, X (Y τ are linearly independent, where τ denotes the transpose of a matrix. Now we prove the claim. Suppose that there exist β and β 2 with β 2 0 such that β (X, Y +β 2 (X (X, X (Y = 0. So we have X (β (X, Y +(β +X (β 2 (X (X, X (Y + β 2 (X 2 (X, X 2 (Y = 0. Since β 2 0, we are in contradiction with the fact that V 2 and V 3 linearly independent. So the claim is proved. Hence by (2 and the linear independency we get that X (α 2 = X (α 3 = 0onZ = 0 and so α 2 and α 3 are constants on Z = 0, denoted by c 2 and c 3 respectively. Hence it follows from ( with l = 0 that + c 2 X + c 3 Y = 0onZ = 0, a contradiction. The case α 0 and V 2 and V 3 linearly dependent can be studied obtaining also a contradiction. Similarly ifs α = 0 then we can get also a contradiction working with either α 2 0or α 3 0. These contradictions show that we cannot work with the multiplicity of Z = 0inCP 2. Acknowledgements The first author is partially supported by a MCYT/FEDER grant number MTM and by a CICYT grant number 2009SGR 40. The second author is partially supported by NNSF of China grant and NCET of China grant He thanks the Centre de Recerca Matemàtica for the hospitality and the financial support by the grant SAB (Ministerio de Educación y Ciencia, Spain. Appendix A. The Poincaré compactification In this appendix we introduce the Poincaré compactification of the vector field X in R n, which wasusedinsection2. Let X be the polynomial vector field in R n of degree d given in (. For convenience we also use (P (x,..., P n (x to denote the vector field X.LetH ={x R n+ : x n+ = } be the hyperplane in R n+ and S n ={x R n+ : x 2 + +x2 n+ = } be the n-sphere in Rn+. Denote by S n + and Sn the half-sphere with x n+ > 0 and x n+ < 0, respectively. We define the epimorphisms given by and φ + : R n S n + and φ : R n S n, φ + (x = (x (x,...,x n, (y,...,y n+, φ (x = (x (x,...,x n, (y,...,y n+, where = + x 2 + +x2 n. These induce a vector field X in Sn + Sn by X (y = (Dφ + x X (x for y = φ + (x and by X (y = (Dφ x X (x for y = φ (x.

13 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( The expression of X on S n + Sn is y 2 y 2 y y 3 y y n y P y y 2 y2 2 y 3 y 2 y n y 2 P 2 X (y = y n+.... y y n y 2 y n y 3 y n yn 2., Pn y y n+ y 2 y n+ y 3 y n+ y n y n+ where Pi = P i (y /y n+,...,y n /y n+ ifs y n+ > 0, or Pi = P i ( y /y n+,..., y n /y n+ if y n+ < 0. This defines an analytic vector field on S n, namely y m n+ X (y. This vector field is called the Poincaré compactification of X and it is denoted by p(x. To obtain the analytic expression for p(x we shall consider the n-sphere as a differentiable manifold. Choose the 2n + 2 coordinate neighborhoods of S n given by U i ={y S n : y i > 0} and V i ={y S n : y i < 0} for i =,...,n+. The corresponding coordinate maps Φ i : U i R n and Ψ i : V i R n are defined by Φ i (y = Ψ i (y = (y j /y i,...,y jn /y i with j < <j n n + and j k i for k =,...,n. We now compute the expression of p(x on U i.inu,lety U S +. Consider the tangent map (DΦ y : T y U T Φ (yr n, we have ( (DΦ y y m+ n+ X (y = y m n+ (DΦ y (Dφ + x X (x = y m n+ D(Φ φ + x X (x, where y = φ + (x. Hence D(Φ φ + x X (x = x 2 ( x 2 P + x P 2,..., x n P + x P n, P. (3 Let (z,...,z n be the coordinates of U,i.e.(z,...,z n = Φ (y,...,y n+. Then (3 becomes D(Φ φ + x X (x = z n ( z P + P 2,..., z n P + P n, z n P, where P i = P i (/z n,z /z n,...,z n /z n. Since y m n+ = (z n/ (z m, the vector field X becomes (z m ( z P + P 2,..., z n P + P n, z n P, (4 where P i = zn mp i(/z n,z /z n,...,z n /z n for i =,...,n. If y U S we have the same expression for p(x as in (4. Working in a similar way we can prove that p(x in U j for j = 2,...,nhas the expression (z m ( z P j + P,..., z j P j + P j, z j P j + P j+,..., z n P j + P n, z n P j, with P i = zn mp i(z /z n,...,z j /z n, /z n,z j+ /z n,...,z n /z n for i =,...,n. The expression for p(x in U n+ is zn m+ (P (z,...,p n (z.

14 778 J. Llibre, X. Zhang / Bull. Sci. math. 33 ( In V i we have y m n+ = ( z n/ (z m = ( m (z n / (z m. So the expression for p(x in V i is the same as in U i multiplied by ( m. The vector field p(x allows us to study the behavior of X in the neighborhood of infinity, that is, in the neighborhood of the equator S n ={y S n : y n+ = 0}. From the expressions of p(x in the local charts U i and V i for i =,...,n+, we deduce that the infinity is invariant by the flow of p(x and that p(x has two copies of X on the northern and southern hemisphere of S n, that is, on U n+ and V n+, respectively. References [] C. Christopher, J. Llibre, J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 ( [2] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges, Bull. Sci. Math. 2ème série 2 ( , 23 44, [3] G. Darboux, De l emploi des solutions particulières algébriques dans l intégration des systèmes d équations différentielles algébriques, C. R. Math. Acad. Sci. Paris 86 ( [4] V.A. Dobrovol skii, N.V. Lokot, J.-M. Strelcyn, Mikhail Nikolaevich Lagutinskii (87 95: un mathématicien méconnu, Historia Math. 25 ( [5] J. Giné, J. Llibre, A family of isochronous foci with Darboux first integral, Pacific J. Math. 28 ( [6] J.P. Jouanolou, Equations de Pfaff algébriques, Lecture Notes in Math., vol. 708, Springer-Verlag, New York, 979. [7] J. Llibre, G. Rodríguez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 98 ( [8] J. Llibre, C. Valls, Integrability of the Bianchi IX system, J. Math. Phys. 46 ( , 3 pp. [9] J. Llibre, C. Valls, On the integrability of the Einstein Yang Mills equations, J. Math. Anal. Appl. 336 ( [0] J. Llibre, X. Zhang, Rational first integrals in the Darboux theory of integrability in C n, Bull. Sci. Math., doi:0.06/j.bulsci , in press. [] J. Llibre, X. Zhang, Darboux theory of integrability in C n taking into account the multiplicity, J. Differential Equations, in press. [2] J.V. Pereira, Vector fields, invariant varieties and linear systems, Ann. Inst. Fourier 5 ( [3] O.E. Rössler, An equation for continuous chaos, Phys. Lett. A 57 ( [4] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 ( [5] D. Schlomiuk, N. Vulpe, Planar quadratic vector fields with invariant lines of total multiplicity at least five, Qual. Theory Dyn. Syst. 5 ( [6] C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A 35 ( [7] X. Zhang, Exponential factors and Darbouxian first integrals of the Lorenz system, J. Math. Phys. 43 ( [8] X. Zhang, Exponential factors and Darbouxian integrability for the Rössler system, Internat. J. Bifur. Chaos 4 (