Polynomial differential systems having a given Darbouxian first integral

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1 Bull. Sci. math Polynomial diffeential systems having a given Dabouxian fist integal Jaume Llibe, Chaa Pantazi Deatament de Matemàtiques, Univesitat Autònoma de Bacelona, Bellatea, Bacelona, Sain Received 4 Ail 2004; acceted 7 Ail 2004 Available online 25 May 2004 Abstact The Dabouxian theoy of integability allows to detemine when a olynomial diffeential system in C 2 has a fist integal of the kind f λ 1 1 f λ exg/h whee f i, g and h ae olynomials in C[x,y], andλ i C fo,...,. The functions of this fom ae called Dabouxian functions. Hee, we solve the invese oblem, i.e. we chaacteize the olynomial vecto fields in C 2 having a given Dabouxian function as a fist integal. On the othe hand, using infomation about the degee of the invaiant algebaic cuves of a olynomial vecto field, we imove the conditions fo the existence of an integating facto in the Dabouxian theoy of integability Elsevie SAS. All ights eseved. MSC: 34C05; 34A34; 34C14 Keywods: Polynomial diffeential system; Fist integal; Dabouxian function 1. Intoduction and statement of the main esults By definition a lana olynomial diffeential system is a diffeential system of the fom dx dt =ẋ = Px,y, dy dt =ẏ = Qx,y, 1 The authos ae atially suoted by a MCYT gant BFM C02-02 and by a CIRIT gant numbe 2001SGR * Coesonding autho. addess: llibe@mat.uab.es J. Llibe /$ see font matte 2004 Elsevie SAS. All ights eseved. doi: /.bulsci

2 776 J. Llibe, Ch. Pantazi / Bull. Sci. math whee P and Q ae olynomials in the vaiables x and y. Moeove, the deendent vaiables x and y, the indeendent vaiable t called the time, and the coefficients of the olynomials P and Q ae comlex. Associated to the olynomial diffeential system 1 in C 2 thee is the olynomial vecto field X = Px,y x + Qx,y 2 y in C 2. Sometimes, the olynomial vecto field X will be denoted simly by P, Q. The degee m of the olynomial diffeential system 1 o of the olynomial vecto field X is the maximum of the degees of the olynomials P and Q. The degee of a olynomial P is denoted by δp. The degee of a ational function P/Q is defined as δp/q = max{δp,δq}. If the olynomials P and Q ae not coime, let R be the geatest common diviso of P and Q. Then, the change in the indeendentvaiable t given by ds = Rdttansfoms the olynomial vecto field 2 into the olynomial vecto field P /R, Q/R with P/R and Q/R coime. Since if P /R, Q/R has a fist integal, we also have a fist integal fo P, Q, in what follows we shall wok with olynomial vecto fields P, Q with P and Q coime. A Dabouxian function can be witten into the fom Hx,y = f λ 1 1 f λ g ex f n 1 1 f n, 3 whee f 1,...,f ae ieducible olynomials in C[x,y], λ 1,...,λ C, n 1,...,n N {0} i.e. the n i ae non-negative integes and the olynomial g of C[x,y] is coime with f i if n i 0. Fist we want to chaacteize when a olynomial vecto field X in C 2 has the Dabouxian function Hx,y as a fist integal; i.e. when H is constant on the taectoies of X contained in the domain of definition U of H, o equivalently when dh/dt = XH = P H/ x+ Q H/ y = 0, on U. Given a olynomial vecto field X the Dabouxian theoy of integability ovides sufficient conditions in ode that X has a Dabouxian fist integal, see fo moe details Section 2. This theoy stated with Daboux [10] in Fo moe details and esults on the Dabouxian theoy of integability fo lana olynomial vecto fields, see [1,3,4,6,12, 14 19]. Hee, we study the invese oblem. Ou main esults on the invese oblem ae summaized in what follows. Theoem 1. Let Hx,y = f λ 1 1 f λ exg/f n 1 1 f n be a Dabouxian function with f 1,...,f ieducible olynomials in C[x,y], λ 1,...,λ C, n 1,...,n N {0} and the olynomial g of C[x,y] is coime with f i if n i 0. We denote by l the degee of the ational function g/f n 1 1 f n. Then, H is a fist integal fo the olynomial vecto field X = P, Q of degee m with P and Q coimes if and only if a l + δf i = m + 1 and

3 X = l=1 g J. Llibe, Ch. Pantazi / Bull. Sci. math f n l l λ i n i =1, i =1, i X f X f + X g, 4 whee X fi is the Hamiltonian vecto field f iy,f ix. Moeove, the vecto field given by 4 has the integating facto R 1 = f 1 f f n 1 1 f n 1. b l + δf i >m+ 1 and X is as in 4 dividing its comonents by thei geatest common diviso D. Moeove, DR 1 is a ational integating facto of X. Theoem 1 will be oved in Section 3. Also in that section we shall show that the second at of statement a cannot be extended to the integating factos of the fom 3 with g 0. In Section 4 we ovide examles of all statements of Theoem 1. Coollay 2. Unde the assumtions of Theoem 1 if 3 is a fist integal fo the olynomial vecto field X = P, Q of degee m with P and Q coimes, then l + δf i m + 1. =1 Coollay 2 follows diectly fom Theoem 1. Note that Coollay 2 says that the degee of a olynomial vecto field having the fist integal 3 is not indeendent of the degees of the olynomials aeaing in 3. Pelle and Singe in [15] oved the following esult. Theoem 3. If a olynomial vecto field X has a fist integal of the fom Hx,y = f λ 1 1 f λ exg/f n 1 1 f n whee f 1,...,f ae ieducible olynomials in C[x,y], λ 1,...,λ C, n 1,...,n N {0} and the olynomial g of C[x,y] is coime with f i if n i 0, then the vecto field has an integating facto of the fom 1 ax,y N bx,y with a,b C[x,y] and N an intege. We imove Theoem 3 as follows. Coollay 4. We assume that the olynomial vecto field X has a fist integal of the fom Hx,y= f λ 1 1 f λ exg/f n 1 1 f n whee f 1,...,f ae ieducible olynomials in C[x,y], λ 1,...,λ C, n 1,...,n N {0} and the olynomial g of C[x,y] is coime with f i if n i 0. We denote by l = δg/f n 1 1 f n. a If l + δf i = m + 1 then the invese of the olynomial f 1 f f n 1 1 f n integating facto. is an

4 778 J. Llibe, Ch. Pantazi / Bull. Sci. math b Othewise, a function of the fom ax,y/f 1 f f n 1 1 f n with a C[x,y] is an integating facto. The esults of Coollay 4 ae stongly elated with Poosition 3.2 and Coollay 3.3 of Walche [20]. Othe asects of the invese oblem of the Dabouxian theoy of integability have been studied, see fo moe details Theoem 10 due to Chistohe [5], Żoł adek [21] and Chistohe, Llibe, Pantazi and Ziang [8] and Poosition 12 due to Chistohe and Kooi [5] in Section 2. In fact, the next esult imoves statement b of Theoem 10 and Poosition 12. Theoem 5. Let X = P, Q be a olynomial vecto field with P and Q coime having f 1 = 0,...,f = 0 as ieducible invaiant algebaic cuves satisfying the geneic conditions: i Thee ae no oints at which f i and its fist deivatives ae all vanish. ii The highest ode tems of f i have no eeated factos. iii If two cuves intesect at a oint in the finite lane, they ae tansvesal at this oint. iv Thee ae no moe than two cuves f i = 0 meeting at any oint in the finite lane. v Thee ae no two cuves having a common facto in the highest ode tems. Then, X has the fist integal f λ 1 1 f λ Moeove, X = with λ i C if and only if δf i = m + 1. λ i X fi. 5 Theoem 5 will be oved in Section 3. An examle of a olynomial vecto field satisfying Theoem 5 will be given in Section 4. A function Rx,y is an integating facto of the vecto field X = P, Q on the domain of definition U of R if divrp, RQ = 0onU.Asusualthedivegence of the vecto field X is defined by divx = divp, Q = P x + Q y. Fo the next theoem see the definitions of ieducible invaiant algebaic cuve, exonential facto, thei cofactos and weak indeendent singula in Section 2. This theoem imoves the conditions fo the existence of an integating facto in the Dabouxian theoy of integability using infomation about the degee of the invaiant algebaic cuves, secifically it imoves statement e of Theoem 9. As fa as we know, this is the fist time that infomation about the degee of the invaiant algebaic cuves, instead of the numbe of these cuves, is used fo studying the integability of a olynomial vecto field.

5 J. Llibe, Ch. Pantazi / Bull. Sci. math Theoem 6. Suose that a olynomial vecto field X = P, Q of degee m, with P and Q coime, admits ieducible invaiant algebaic cuves f i = 0 with cofactos K i fo,...,; q exonential factos exg /h with cofactos L fo,...,q; and indeendent singula oints x k,y k such that f i x k,y k 0 fo,..., and fo k = 1,...,. Then, the ieducible factos of the olynomials h ae some f i s and we can wite ex g1 h 1 µ1 ex gq h q µq µ1 g 1 = ex = ex h µ qg q g f n 1 1 f n whee µ 1,...,µ q C, n 1,...,n N {0} and the olynomial g of C[x,y] is coime with f i if n i 0. We denote by l = max{ n iδf i,δg}. If + q + = mm + 1/2, l + δf i <m+ 1, and the indeendent singula oints ae weak, then the multi-valued function µ1 µq f λ 1 1 f λ g1 gq ex ex 6 h 1 fo convenient λ i,µ C not all zeo is an integating facto of X. h q Theoem 6 is also oved in Section 3. An examle of a olynomial vecto field satisfying Theoem 6 will be given in Section 4. As fa as we know, this theoem uses by fist time infomation about the degee of the invaiant algebaic cuves fo studying the integability of a olynomial vecto field, because until now the Dabouxian theoy of integability only used of the invaiant algebaic cuves of a olynomial vecto field its numbe fo studying its integability looking fo, eithe a fist integal, o an integating facto, see Theoem 9., h q 2. Dabouxian theoy of integability The Dabouxian theoy of integability fo lana olynomial vecto fields can be summaized in the next theoem. As fa as we know, the oblem of integating a olynomial vecto fields by using its invaiant algebaic cuves was stated to be consideed by Daboux in [10]. The vesion that we esent imoves Daboux s one essentially because hee we also take into account the exonential factos see [4,9], and the indeendent singula oints see [3]. Some moe comlete vesions can also conside the Dabouxian invaiants see [1,2], but since these moe comlete vesions will not lay any ole in this ae hee we omit them. Fist we intoduce the main thee notions in the Dabouxian theoy of integability. Let f C[x,y]. The algebaic cuve fx,y = 0isaninvaiant algebaic cuve of the olynomial vecto field X if fo some olynomial K C[x,y] we have Xf = P f x + Q f y = Kf.

6 780 J. Llibe, Ch. Pantazi / Bull. Sci. math The olynomial K is called the cofacto of the invaiant algebaic cuve f = 0. Of couse, the cuve f = 0 is fomed by taectoies of the olynomial vecto field X. We note that since the olynomial vecto field has degee m, then any cofacto has at most degee m 1. The following esult is well known, see fo instance [7]. Poosition 7. We suose that f C[x,y] and let f = f n 1 1 f n be its factoization in ieducible factos ove C[x,y]. Then, fo the olynomial system 1, f = 0 is an invaiant algebaic cuve with cofacto K f if and only if f i = 0 is an invaiant algebaic cuve fo each,..., with cofacto K fi. Moeove K f = n 1 K f1 + +n K f. By Poosition 7, in what follows we can estict ou attention to the ieducible invaiant algebaic cuves. Let h, g C[x,y] and assume that h and g ae elatively ime in the ing C[x,y].Then the function exg/h is called an exonential facto of the olynomial vecto field X if fo some olynomial L C[x,y] of degee at most m 1 it satisfies g g X ex = L ex. h h As befoe we say that L is the cofacto of the exonential facto exg/h. Poosition 8. If exg/h is an exonential facto fo the olynomial vecto field X,then h = 0 is an invaiant algebaic cuve of X. Poof. See [4]. In fact, in Poosition 8 h = 0 is an invaiant algebaic cuve with multilicity lage than 1 as solution of X, fo moe details see [9]. If Sx,y = m 1 i+=0 a i x i y is a olynomial of degee m 1 with mm + 1/2 coefficients in C, then we wite S C m 1 [x,y]. We identify the linea vecto sace C m 1 [x,y] with C mm+1/2 though the isomohism S a 00,a 10,a 01,...,a m 1,0,a m 2,1,...,a 0,m 1. We say that oints x k,y k C 2, k = 1,...,, ae indeendent with esect to C m 1 [x,y] if the intesection of the hyelanes { a i C mm+1/2 : m 1 i+=0 } xk i y k a i = 0, k= 1,...,, is a linea subsace of C mm+1/2 of dimension mm + 1/2 >0. We ecall that x 0,y 0 is a singula oint of system 1 if Px 0,y 0 = Qx 0,y 0 = 0. We emak that the maximum numbe of isolated singula oints of the olynomial system 1 is m 2 by Bezout theoem, that the maximum numbe of indeendent isolated singula oints of the system is mm + 1/2, and that mm + 1/2 <m 2 fo m 2.

7 J. Llibe, Ch. Pantazi / Bull. Sci. math A singula oint x 0,y 0 of system 1 is called weak if the divegence, div X,ofsystem 1 at x 0,y 0 is zeo. Theoem 9. Suose that a olynomial vecto field X of degee m admits ieducible invaiant algebaic cuves f i = 0 with cofactos K i fo,...,; q exonential factos exg /h with cofactos L fo,...,q; and indeendent singula oints x k,y k such that f i x k,y k 0 fo,..., and fo k = 1,...,. Moeove, the ieducible factos of the olynomials h ae some f i s. a Thee exist λ i,µ C not all zeo such that q λ i K i + µ L = 0, =1 if and only if the multi-valued function µ1 f λ 1 1 f λ g1 ex ex h 1 gq h q µq 7 is a fist integal of X. b If + q + =[mm + 1/2]+1, then thee exist λ i,µ C not all zeo such that q λ i K i + µ L = 0. =1 c If + q + [mm + 1/2]+2,thenX has a ational fist integal, and consequently all taectoies of the system ae contained in invaiant algebaic cuves. d Thee exist λ i,µ C not all zeo such that q λ i K i + µ L = divx, =1 if and only if the function 7 is an integating facto of X. e If + q + = mm + 1/2 and the indeendent singula oints ae weak, then function 7 fo convenient λ i,µ C not all zeo is a fist integal o an integating facto. Note that in Theoem 9 the fact that the ieducible factos of the olynomials h ae some f i s is due to Poosition 8. Statements a, b, d and e esticted only to invaiant algebaic cuves ae due essentially to Daboux [10]. These statements taking into account the exonential factos and the indeendent singula oints can be found in [4,6,7]. Statement c is due to Jouanolou [12], fo an easy oof see [7]. The next theoem is anothe kind of invese oblem of the Dabouxian theoy of integability, in it the invaiant algebaic cuves ae given and we want to obtain all the olynomial vecto fields having these invaiant algebaic cuves. This theoem was stated by Chistohe without oof in [5], and used in othe aes as [1,11,13]. Żoł adek in [21]

8 782 J. Llibe, Ch. Pantazi / Bull. Sci. math see also Theoem 3 of [22] stated a simila esult using an analytic aoach, but as fa as we know the ae [21] has not been ublished. A fist comlete oof of it, using mainly algebaic tools, has been given in [8]. Theoem 10. Let f i = 0, fo,...,, be ieducible algebaic cuves in C 2.We assume that all f i satisfy the geneic conditions of Theoem 5. Then any olynomial vecto field X of degee m having all f i = 0 as invaiant algebaic cuves satisfies one of the following statements. a If δf i <m+ 1,then X = f i Y + h i X fi, whee the h i ae olynomials such that δh i m+1 δf i, and Y is a olynomial vecto field with degee m δf i. b If δf i = m + 1,thenX is of the fom 5. c If δf i >m+ 1,thenX = 0. In [8] we show that all the assumtions of Theoem 10 ae necessay in ode that the esult hold. Moe secifically, we oved the next esult. Poosition 11. If one of the conditions i v of Theoem 10 is not satisfied, then its statements do not hold. An inteesting comlement to Theoem 10b due to Chistothe and Kooi [5] is the following. Poosition 12. Unde the assumtions of Theoem 10b a olynomial system 5 has an integating facto of the fom f 1 f 1 and a fist integal of the fom f λ 1 1 f λ. The second at of statement a of Theoem 1 is in some sense the equivalent to Poosition 12 fo ou invese oblem. 3. Poof of ou main esults In this section we shall ove Theoems 1, 5 and 6. Poof of Theoem 1. By a diect calculation we ove that system 4 in statements a and b of Theoem 1 has 3 as a fist integal. So, the only if at of Theoem 1 is oved. Now, we shall ove the if at.

9 J. Llibe, Ch. Pantazi / Bull. Sci. math We assume that H = f λ 1 1 f λ F with F = exg/f n 1 1 f n is a fist integal of the olynomial vecto field X = P, Q of degee m. So, we have 0 = PH x + QH y = PF λ i f λ i 1 i f ix + g x f n g + QF g n i f n i 1 i f ix [ = P λ i f ix g λ i f λ i 1 i f iy n i f n i 1 i f iy n i f ix + Q λ i f iy g = 1 n i f iy = 1 f λ f n f n f λ + g x f n =1 + g y = 1 + g y f n = 1 =1 f n = 1 f 2n = 1 f 2n f n ]F f n f λ f λ f λ f λ 1. Since the last exession is equal to zeo, we can cancel the non-zeo oduct F =1 f λ 1 and we can elace it with the non-zeo oduct =1 f n. So we get 0 = PG 1 + QG 2, 8 with G 1 = λ i f ix G 2 = λ i f iy =1 = 1 f n f n + g x + g y g g n i f ix n i f iy f λ,.

10 784 J. Llibe, Ch. Pantazi / Bull. Sci. math We emak that, since P and Q ae coime, fom PH x + QH y = 0 it follows that H x and H y cannot be zeo. Consequently, G 1 and G 2 ae not zeo. Since P and Q ae coime, fom 8 we have that P must divide the olynomial G 2, and Q must divide the olynomial G 1, which is imossible if δg i <m= max{δp,δq} fo, 2. Due to the fact that δg i = l 1 + δf i, we get that l + δf i m + 1. Since P and Q ae coime, if δf i + l = m + 1 we have that thee is a constant λ C \{0} such that P = λg 2 and Q = λg 1. Doing the change of time t 1/λt the fist at of statement a is oved. Now we shall show the second at of statement a. The algebaic cuve f k = 0 is invaiant fo the vecto field 4 with cofacto K k = λ i f ix f ky f iy f kx l = 1 f n l l + g x f ky g y f kx The vecto field 4 has divegence div X = o equivalently, div X = l = 1 + l = 1 f n l l f n l l x k + g, k n i f iy f kx f ix f ky λ i f iy + y l = 1 f n l l y λ i f ix f iy + g x n i f iy g y x n i f ix + g n i f iy f ix + i, k = 1 y g x n k λ i f ky f ix f kx f iy x g y, y l = 1 l k f n l l, k. λ i f ix x f n k 1 k

11 + + + g + J. Llibe, Ch. Pantazi / Bull. Sci. math l = 1 f n l l,k = 1 λ i y f ix x f iy n i g x f iy g y f ix n i x f iy y f ix g x f iy g y f ix, and it is easy to check that K + n K = div X. = 1 = 1 k = 1 k i, k = 1 k i, Theefoe, by Theoem 9b, R 1 = f 1 f f n 1 1 f n 1 is an integating facto of the vecto field 4. Suose that l + δf i >m+ 1. Since P and Q ae coime, fom 8 we have that thee is a olynomial F such that G 1 = FQ and G 2 = FP. So, dividing G 1 and G 2 by F we obtain the olynomial vecto field P, Q of degee m. This comletes the oof of statement b, and consequently of Theoem 1. Poof of Theoem 5. Assume that the assumtions of Theoem 5 hold. Suose that δf i = m + 1. Then, by Theoem 10b it follows that the olynomial vecto field satisfying the assumtions of Theoem 5 is of the fom 5, and by Poosition 12 it has the fist integal f λ 1 1 f λ. Now we shall ove the convese statement. Suose that the olynomial vecto field satisfying the assumtions of Theoem 5 has the fist integal f λ 1 1 f λ. So, fo this fist integal l = 0, using the notation of Theoem 1. Then, by Coollay 2 we have that δf i m + 1. Since all the invaiant algebaic cuves f i = 0 ae geneic, by Theoem 10, it follows that δf i m + 1. Hence, δf i = m + 1, and the oof of the theoem is comleted. Now we shall show that the second at of statement a of Theoem 1 cannot be extended to integating factos of the fom 3 with g 0. The system ẋ = xx + y + 1, 9 ẏ = yx + y, has the two invaiant algebaic cuves f 1 = x = 0andf 2 = y = 0, and the exonential facto F = ex 1 + x/y with cofactos K 1 = x + y + 1, K 2 = x + y and L = 1, f k f k

12 786 J. Llibe, Ch. Pantazi / Bull. Sci. math esectively. Since K 1 + K 2 + L = 0, by Theoem 9a system 9 has the fist integal H = f 1 1 f 2 F. Doing simle comutations we obseve that system 9 can be witten into the fom 4 with λ 1 = 1, λ 2 = 1, n 1 = 0andn 2 = 1. We also note that the olynomials P and Q ae coime. Since the divegence of system 9 is div = 1 +3x +3y and we have that K 1 +K 2 div and K 1 + K 2 + L div, by Theoem 9d thee is no integating factos of the fom f 1 f 2 1 o f 1 f 2 ex F 1. So, although system 9 can be witten into the fom 4, the second at of statements a of Theoem 1 cannot be extended to integating factos of the fom 3 with g 0. Howeve, since K 1 + 2K 2 = div, this system has the integating facto R 1 = f1 1 f2 2. Poof of Theoem 6. Assume that the assumtions of Theoem 6 hold. By Theoem 9e function 6 is eithe a fist integal, o an integating facto of X. But, fom Coollay 2 function 6 cannot be a fist integal of X because l + δf i <m+ 1. Hence, the oof is comleted. 4. The examles Fist, we ovide thee examles of a fist integal satisfying statement a of Theoem 3. The Dabouxian function H = y 3 ex3x 3 /y is of the fom 3 with f 1 = y, λ 1 = 3, n 1 = 1andg = 3x 3. Then, the l defined in Theoem 3 satisfies l = 3. Theefoe, since l + δf i = 4, and the olynomial vecto field given by 4 is X = 3y + x 3, 3x 2 y with m = 3, it follows that H and X satisfy statement a of Theoem 3. The next fist integal and its coesonding olynomial vecto field ovide examles satisfying Theoem 3a and Theoem 5. The Dabouxian function H = xyx 1 + y/3 is of the fom 3 with f 1 = x, f 2 = y, f 3 = x 1+y/3, λ 1 = λ 2 = λ 3 = 1, n 1 = n 2 = n 3 = 0 and g = 0. Then, the l = 0. Theefoe, since l + δf i = 3, and the olynomial vecto field given by 4 is X = x1 x 2y/3, y 1 + 2x + y/3 with m = 2, we get that H and X satisfy statement a of Theoem 3, because X has the fist integal H and the integating facto 1/H. Additionally, this is an examlesatisfying Theoem5. Now the thid examle satisfying Theoem 3a. The Dabouxian function H = x 2 + y 2 ex2y is of the fom 3 with f 1 = x + iy, f 2 = x iy, λ 1 = λ 2 = 1, n 1 = n 2 = 0 and g = 2y. Then, the l = 1. Theefoe, since l + δf i = 3, and the olynomial vecto fieldgivenby4isx = 2 y x 2 y 2,x with m = 2, we have that H and X satisfy statement a of Theoem 3, because X has the fist integal H and the integating facto 1/x 2 + y 2. Now we shall ovide two examles satisfying statement b of Theoem 3. The Dabouxian function H = y 4 x 3 +x 4 +y 4 is of the fom 3 with f 1 = y, f 2 = x 3 +x 4 + y 4, λ 1 = 4, λ 2 = 1, n 1 = n 2 = 0andg = 0. Then, the l = 0. Theefoe, l + δf i = 5, and the olynomial vecto field given by 4 is P, Q = 4x x,x xy with P and Q non-coime. So, X = 4x1 + x,y3 + 4x with m = 2 is the olynomial vecto field satisfying statement b of Theoem 3. The second examle is the following one. The Dabouxian function H = x y x 2 ex 1/x + 1 is of the fom 3 with f 1 = x + 1, f 2 = y x 2, λ 1 = 2, λ 2 = 1,

13 J. Llibe, Ch. Pantazi / Bull. Sci. math n 1 = 1, n 2 = 0andg = 1. Then, l = 1. Theefoe, l + δf i = 4, and the olynomial vecto field given by 4 is X = x + 1 2, 2x y 3x 2 2xy with m = 2 satisfying statement b of Theoem 3. Finally we ovide an examle satisfying Theoem 6. The olynomial vecto field X = xy + 1, yx + 1 with m = 2 has the invaiant algebaic cuve f 1 = x with cofacto K 1 = y +1, the exonential facto exx +y +1 with cofacto L = x y = div, and the weak indeendent singula oint 1, 1 which is not on f 1 = 0. Theefoe, l = 1, = q = = 1, and consequently it satisfies + q + = mm + 1/2 = 3and l + δf i = 2 <m+ 1 = 3, and it has f1 0 exx + y + 1 as integating facto. Hence, X is an examle of a olynomial vecto field satisfying Theoem 6. We note that, fom Theoem 9a, thee does not exist a fist integal given by a Dabouxian function of the fom f λ 1 1 exx + y + 1µ 1. Refeences [1] L. Caió, M.R. Feix, J. Llibe, Integability and algebaic solutions fo lana olynomial diffeential systems with emhasis on the quadatic systems, Resenhas de Univesidade de São Paulo [2] L. Caió, J. Llibe, Dabouxian fist integals and invaiants fo eal quadatic systems having an invaiant conic, J. Phys. A [3] J. Chavaiga, J. Llibe, J. Sotomayo, Algebaic solutions fo olynomial systems with emhasis in the quadatic case, Exositiones Math [4] C.J. Chistohe, Invaiant algebaic cuves and conditions fo a cente, Poc. Roy. Soc. Edinbugh A [5] C. Chistohe, R. Kooi, Algebaic invaiant cuves and the integability of olynomial systems, Al. Math. Lett [6] C. Chistohe, J. Llibe, Algebaic asects of integability fo olynomial systems, Qualitative Theoy of Plana Diffeential Equations [7] C. Chistohe, J. Llibe, Integability via invaiant algebaic cuves fo lana olynomial diffeential systems, Ann. Diffeential Equations [8] C.J. Chistohe, J. Llibe, C. Pantazi, X. Ziang, Daboux integability and invaiant algebaic cuves fo lana olynomial systems, J. Phys. A: Gen. Math [9] C. Chistohe, J. Llibe, J.V. Peeia, Multilicity of invaiant algebaic cuves in olynomial vecto fields, Peint, [10] G. Daboux, Mémoie su les équations difféentielles algébiques du emie ode et du emie degé Mélanges, Bull. Sci. Math. 2ème séie ; ; [11] R.E. Kooi, Real olynomial systems of degee n with n + 1 line invaiants, J. Diffeential Equations [12] J.P. Jouanolou, Equations de Pfaff algébiques, in: Lectues Notes in Mathematics, vol. 708, Singe-Velag, New Yok, [13] J. Llibe, J.S. Péez del Río, J.A. Rodíguez, Phase otaits of a new class of integable quadatic vecto fields, Dynamics of Continuous, Discete and Imulsive Systems [14] J.V. Peeia, Integabilidade de equaçoes difeenciais no lano comlexo, in: Monogafias del IMCA, vol. 25, Lima, Peu, [15] M.J. Pelle, M.F. Singe, Elementay fist integals of diffeential equations, Tans. Ame. Math. Soc [16] D. Schlomiuk, Elementay fist integals and algebaic invaiant cuves of diffeential equations, Exositiones Math [17] D. Schlomiuk, Algebaic aticula integals, integability and the oblem of the cente, Tans. Ame. Math. Soc

14 788 J. Llibe, Ch. Pantazi / Bull. Sci. math [18] D. Schlomiuk, Algebaic and geometic asects of the theoy of olynomial vecto fields, in: D. Schlomiuk Ed., Bifucations and Peiodic Obits of Vecto Fields, 1993, [19] M.F. Singe, Liouvillian fist integals of diffeential equations, Tans. Ame. Math. Soc [20] S. Walche, Plane olynomial vecto fields with escibed invaiant cuves, Poc. Roy. Soc. Edinbugh A [21] H. Żoł adek, The solution of the cente-focus oblem, Peint, [22] H. Żoł adek, On algebaic solutions of algebaic Pfaff equations, Studia Math

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