ISOCHRONICITY FOR TRIVIAL QUINTIC AND SEPTIC PLANAR POLYNOMIAL HAMILTONIAN SYSTEMS

ISOCHRONICITY FOR TRIVIAL QUINTIC AND SEPTIC PLANAR POLYNOMIAL HAMILTONIAN SYSTEMS FRANCISCO BRAUN JAUME LLIBRE AND ANA C. MEREU Abstract. In this paper we completely characterize trivial polynomial Hamiltonian isochronous centers of degrees 5 and 7. Precisely we provide simple formulas up to linear change of coordinates for the Hamiltonians of the form H = f 1 +f / where f = f1f : R R is a polynomial map with detdf = 1 f00 = 00 and the degree of f is 3 or 4. 1. Introduction Let Pxy and Qxy be real polynomials in the variables x and y. We say that a polynomial vector field X = PQ has degree n when max{degpdegq} = n. Given a polynomial Hamiltonian H : R R of degree n + 1 the associated polynomial Hamiltonian system of degree n is 1 ẋ = H y xy ẏ = H x xy. System 1 has a center at 00 if there is a neighbourhood of the origin filled of periodic orbits except the origin. The maximum connected set filled of periodic orbits having in its inner boundary the origin is called the period annulus of the center localized at the origin. If the period annulus is R \{00} wecall thecenter global. We say that apolynomial Hamiltonian system has an isochronous center at the origin if 00 is a center of 1 and all the orbits in the period annulus of the center have the same period. The following characterization of the polynomial Hamiltonian systems possessing an isochronous center at the origin was given in [6]. The polynomial Hamiltonian system 1 has an isochronous center of period π at the origin if and only if Hxy = f 1xy +f xy forallxyinaneighborhoodn 0 oftheorigin wheref = f 1 f : N 0 R is an analytic map with Jacobian determinant det Df constant and equal to 1 and f00 = 00. We observe that this characterization still holds 010 Mathematics Subject Classification. Primary 34A34 ; Secondary: 34C5 37C37 14R15. Key words and phrases. Isochronous centers polynomial Hamiltonian systems Jacobian conjecture. 1

F. BRAUN J. LLIBRE AND A. C. MEREU for analytic Hamiltonians. When f can be taken polynomial we say that the polynomial Hamiltonian isochronous center is trivial. In this case it is clear that f will be defined in all R. From [8] when the center is trivial it is a global center if and only if f is globally injective. Thus the problem of knowing whether a trivial polynomial Hamiltonian isochronous center is global or not is equivalent to the Jacobian conjecture in R which stands that a polynomial map f : R R with constant Jacobian determinant is globally injective. We mention here that if the degrees of f 1 and f are less than or equal to 101 then f is globally injective see [7] and for other results on the Jacobian conjecture see [3]. Thus all the trivial polynomial Hamiltonian isochronous centers of degree less than or equal to 01 are global ones. In[] it was proved that all isochronous centers of cubic polynomial Hamiltonian systems are trivial and hence global and after a linear change of coordinates the Hamiltonian can be written as Hxy = k 1 x + k y +k 3 x+k 4 x where k 1 k k 3 k 4 R and k 1 k 0. We also mention that there are no polynomial Hamiltonian isochronous centers of degree 4 see [4]. Moreover it was proved in [5] that there are no polynomial Hamiltonian isochronous centers of even degree for which the analytical function f of the Hamiltonian is defined in the whole plane. In particular there are no trivial polynomial Hamiltonian isochronous centers of even degree. On the other hand there are examples of non-trivial polynomial Hamiltonian isochronous centers of degree 6k+1 for all k 1 see section 3. We point out that in these examples the map f is defined in the whole plane. The following are thus natural questions. Open question 1: Are there non-trivial quintic polynomial Hamiltonian isochronous centers? Open question : Are there non-trivial polynomial Hamiltonian isochronous centers with Hamiltonian such that f is not analytical in the whole plane R? We observe that if the open question has a negative answer then by [5] there are no polynomial Hamiltonian isochronous centers of even degree. Our main result is the characterization of the quintic and the septic trivial polynomial Hamiltonian isochronous centers see theorems 4 and 5 respectively where we provide formulas for the Hamiltonian of these systems. We also give an alternative formula for the Hamiltonian of the cubic polynomial isochronous centers with a trivial proof using that these centers are trivial see Proposition 3.. Trivial polynomial Hamiltonian isochronous centers We shall use the following technical result.

ISOCHRONICITY FOR QUINTIC AND SEPTIC HAMILTONIAN SYSTEMS 3 Lemma 1. Let pq : R R be homogeneous polynomials of degree m and n respectively such that detdpq 0. Let also d = gcdmn we define m = m/d and n = n/d. Then there exists a homogeneous polynomial r : R R of degree d and constants c p c q R such that p = c p r m and q = c q r n. Proof. Itisenoughtoprovethattherationalfunctionf = p n /q m isconstant. In order to do that it is enough to show that f x = f y = 0. We have f x = pn 1 q m 1 q m nqp x mpq x = pn 1 q m 1 q m yp x q y q x p y = 0 where in the second equality above we used the Euler s Theorem for homogeneous maps. The proof that f y = 0 is analogous. We address the reader to [1] for a much more general version of Lemma 1. In the proofs of the results of this section we will have to solve partial differential equations of the form p x +βp y = h where ph : R R are homogeneous polynomials of degrees k and k 1 respectively and β R. By defining q : R R by 3 pxy = qxy βx = qx 1 y 1 the original equation turns to q x1 = hx 1 y 1 +βx 1 and hence there is c R such that qx 1 y 1 = x 1 0 hsy 1 +βsds+cy k 1 and p is given by 3. For further references we enunciate this procedure in the following result. Lemma. Let ph : R R be homogeneous polynomials of degrees k and k 1 respectively satisfying p x +βp y = h with β R. Then there is c R such that pxy = qxy βx where qx 1 y 1 = x 1 0 hsy 1 +βsds+cy k 1. We begin with Proposition 3 where we give an alternative characterization of the cubic polynomial Hamiltonian isochronous centers using the fact that they are trivial recall the mentioned result of []. Right after the proof we relate the formula of Proposition 3 and the one presented in []. Proposition 3. Assume that the polynomial Hamiltonian system 1 has degree 3 and has an isochronous center of period π at the origin. Then up to a linear change of variables the Hamiltonian can be written as H = P +y +λp with P = x+c y c λ R and c 0.

4 F. BRAUN J. LLIBRE AND A. C. MEREU Proof. Since the cubic polynomial Hamiltonian isochronous centers are trivial there exists f = f 1 f : R R a polynomial map of degree with detdf = 1 and f00 = 00 such that H = f 1 +f /. After a linear change of variables it is clear that we can write f 1 = x+p f = y +q with p and q homogeneous polynomials of degree. Without loss of generality we can assume that p 0 otherwise we change the roles of f 1 and f and of x and y. The hypothesis detdf = 1 also gives that 4 p x +q y = 0 p x q y q x p y = 0. The second equation of 4 gives from Lemma 1 that there exists λ 0 such that q = λp. Substituting this in the first equation of 4 we obtain p x +λp y = 0 which solved for a homogeneous polynomial of degree by Lemma determines c R such that p = c y λx with c 0. We then apply the change of variables xy xy λx finishing the proof. We observe that changing x to y and taking k 1 = 1/ 1+λ k = 1+λ k 3 = λ/ 1+λ and k 4 = c 1+λ theformulaofproposition 3 satisfies the mentioned formula for the cubic polynomial Hamiltonian isochronous centers of []. On the other hand the change of coordinates xy k y +k 3 xk 1 x transforms the mentioned formula of [] in Hxy = x+c y +y / with c = k 4 / k 1 which is the formula of Proposition 3 with λ = 0. We observe that in [] it was not assumed that the isochronous center has period exactly π. For trivial polynomial Hamiltonian isochronous centers of degrees 5 and 7 we have similar formulas to the Hamiltonians see the following theorems 4 and 5. Theorem 4. Assume that the polynomial Hamiltonian system 1 has degree 5 and has a trivial isochronous center of period π at the origin. Then up to a linear change of variables the Hamiltonian can be written as H = P +y +λp with P = x+c y +c 3 y 3 c c 3 λ R and c 3 0. Proof. We consider H = f 1 +f / with f = f1 f a polynomial map of degree 3 satisfyingf00 = 00 and detdfxy = 1 for all xy R. It is clear that after a linear change of variables we can write f 1 = x+p +p 3 f = y +q +q 3

ISOCHRONICITY FOR QUINTIC AND SEPTIC HAMILTONIAN SYSTEMS 5 where p i and q i are homogeneous polynomials of degree i i = 3. Without loss of generality we can suppose that p 3 0. The assumption detdf = 1 gives p x +q y = 0 5 p 3x +q 3y +p x q y q x p y = 0 p 3x q y q x p 3y +p x q 3y q 3x p y = 0 p 3x q 3y q 3x p 3y = 0. The last equation gives by Lemma 1 that there exists λ R such that 6 q 3 = λp 3. Substituting this in the third equation in 5 we obtain 7 p 3x q λp y q λp x p 3y = 0. Here we have two possibilities: either q λp or q = λp. Inthefirstone equation7givesbylemma1thatthereexistabc c 3 R with c c 3 a +b 0 such that 8 q = λp +c ax+by p 3 = c 3 ax+by 3. Using then the first equation of 8 and the first one of 5 we obtain p x +λp y = bc ax+by. From Lemma we obtain d R such that p = bc a+bλx+by λxx+d y λx. Then we substitute p 8 and 6 in the second equation of 5 and after dividing by ax+by and equating the coefficients of x and y to 0 we obtain the system a λ b 1 3a+bλc 3 4 b 3 c +a+bλd c 0 = 0 If a + bλ = 0 it follows that b = 0 since c 0 and hence a = 0 a contradiction. On the other hand if a+bλ 0 it follows that c 3 = 0 which is again a contradiction. If now q = λp it follows from 6 and from the first and the second equations in 5 that p x +λp y = 0 p 3x +λq 3y = 0. Solving these equations for homogeneous polynomials of degrees and 3 see Lemma respectively we obtain c c 3 R such that with c 3 0 because p 3 0. So p = c y λx p 3 = c 3 y λx 3 f 1 = x+c y λx +c 3 y λx 3 f = y +λ c y λx +c 3 y λx 3..

6 F. BRAUN J. LLIBRE AND A. C. MEREU Applying the linear change of variables xy xy λx we end the proof of the theorem. The following is a characterization of trivial polynomial Hamiltonian isochronous centers of degree 7. Theorem 5. Assume that the polynomial Hamiltonian system 1 has degree 7 and has a trivial isochronous center of period π at the origin. Then up to a linear change of variables the Hamiltonian H has one of the following two forms: H = P 1 +y +λp 1 P 1 = x+c y +c 3 y 3 +c 4 y 4 H = P +Q+λP P = x+β 1 Q+β Q where Q = y +Γx c c 3 c 4 β 1 β λγ R and c 4 β Γ 0. Proof. Let f 1 and f be polynomials of degree 4 such that detdf 1 f = 1 and f 1 00 = f 00 = 0. After a linear change of variables it is clear we can write 9 f 1 = x+p +p 3 +p 4 f = y +q +q 3 +q 4 where p i and q i are homogeneous polynomials of degree i for i = 34. Moreover since the homogeneous terms of positive degrees of the Jacobian determinant of f 1 f are zero we obtain the following equations: 10 11 1 13 14 15 p x +q y = 0 p 3x +q 3y +p x q y q x p y = 0 p 4x +q 4y +p x q 3y q 3x p y +p 3x q y q x p 3y = 0 p x q 4y q 4x p y +p 4x q y q x p 4y +p 3x q 3y q 3x p 3y = 0 Equation 15 and Lemma 1 give that 16 q 4 = λp 4. Substituting this in equation 14 yields We have two possibilities either p 3x q 4y q 4x p 3y +p 4x q 3y q 3x p 4y = 0 p 4x q 4y q 4x p 4y = 0. p 4x q 3 λp 3 y q 3 λp 3 x p 4y = 0. 17 q 3 = λp 3 or from Lemma 1 there exist abc 3 c 4 R such that 18 p 4 = c 4 ax+by 4 q 3 = λp 3 +c 3 ax+by 3 with a +b c 3 c 4 0.

ISOCHRONICITY FOR QUINTIC AND SEPTIC HAMILTONIAN SYSTEMS 7 Assuming 17 we obtain from 13 that p 4x q λp y p 4y q λp x = 0. We have then another two possibilities either 19 q = λp or from Lemma 1 there exist abcc c 4 R such that 0 p 4 = c 4 ax +bxy +cy q = λp +c ax +bxy +cy with a +b +c c c 4 0. Assuming 19 it follows from 16 and 17 that equations 10 11 and 1 turn to p x +λp y = 0 p 3x +λp 3y = 0 p 4x +λp 4y = 0 respectively. Solving these equations for homogeneous polynomials of degrees 3 and 4 we obtain that p = c y λx p 3 = c 3 y λx 3 p 4 = c 4 y λx 4. By applying the linear change of coordinates xy xy λx in 9 we obtain the first Hamiltonian of the theorem. Now if we assume 17 and 0 equation 10 turns to p x +λp y +c bx+cy = 0. Using Lemma we obtain d R such that 1 p = c bx+cy +cy λx x+d y λx. Substituting this in 11 using 17 and 0 we obtain a partial differential equation of the form p 3x +λp 3y = h with hxy +λx = 4b acc x +4c bcc +ad +bd λ+lλxy +4c Ly where L = c c +d b+cλ. Then from Lemma we obtain d 3 R such that 3 p 3 = 4 3 c b acx 3 +c bcc +ad +bd λ+lλx y λx +4c Lxy λx +d 3 y λx 3. We finally substitute 1 and 3 and also 16 17 and 0 in equation 1 and after dividing it by we obtain the following homogeneous

8 F. BRAUN J. LLIBRE AND A. C. MEREU identity: 0 = a+bλ ac 4 3c d 3 λ +c 3Lλ bcc d a+3bλ +4b 3 c 3 x 3 + c 4 3ab+λb +ac+3c d 3 λa+λb cλ 4 +c d a+bλb 3cλ+3bcc Lλb cλ ac c +L x y + c 4 3bb+cλ+ac b 3c d 3 a bλ cλ 4c b acd +3cλL xy + cc 4 3c d 3 b+cλ+4cc L y 3 Recalling that a +b +c c c 4 0 we will divide the analysis in two cases: either b = cλ or b+cλ 0. Inthefirstpossibility thecoefficient ofy 3 in4 is4cc L. SinceL = c c it follows that c = 0 and hence L = 0 and b = 0. Then the coefficient of xy turns to 3ac d 3 which gives that d 3 = 0 and hence the coefficient of x 3 gives that c 4 = c d. In particular d 0. With these information it follows from 1 3 and 0 that p = d y λx p 3 = ac d x y λx p 4 = a c d x 4 and q q 3 and q 4 are given by 0 17 and 16 respectively. Then by applying the change of coordinates xy xy λx in 9 it follows that in the new variables f 1 = x+d y +ac x y +a c x 4 f = y +λ x+d y +ac x y +a c x 4 +ac x. By defining β = d Γ = ac and Q = y+γx we clearly obtain the second Hamiltonian of the theorem with β 1 = 0. Now we analyze the second possibility b+cλ 0. The coefficients of y 3 and xy in 4 give the following linear system recall that L is given by c4 4cc 5 A = L d 3 b acd +3cλL where A = 4c cb+cλ 3c b+cλ 3bb+cλ+ac b 3c a bλ cλ The determinant of A is 1c b+cλ 3 0. Thus c 4 and d 3 are given by inverting A in 5. We substitute them in the coefficients of x 3 and x y of 4 and obtain respectively using that c 3 b ac b +ac+3bcλ b+cλ 3 = 0. 6cc 3 b ac b+cλ = 0

ISOCHRONICITY FOR QUINTIC AND SEPTIC HAMILTONIAN SYSTEMS 9 which gives that c 0 and a = b /c. We finally obtain a = b c c 4 = c L b+cλ d 3 = cc L b+cλ with Lcb+cλ 0. Now from 3 and from 0 after some calculations 6 p 3 = c L cb+cλ bx+cy y λx p 4 = c L c b+cλ bx+cy4 q = λp + c c bx+cy. We consider the change of variables xy bx + cy/b + cλy λx whose inverse is the transformation xy x cy/b+cλλx+by/b+ cλ. Observe that the determinant of the change is 1. By applying the transformation in 1 and in 6 we obtain that 7 Therefore 8 p = c b+cλx + L b+cλ y p 3 = c b+cλl x y c p 4 = c b+cλ3 L c x 4 q = λp + c b+cλ x. c p +p 3 +p 4 = c b+cλx + L b+cλ = c b+cλx +β Q y + c b+cλ x c with β = L/b+cλ and Q = y+γx where Γ = c b+cλ /c. Then from 9 in the new variables f 1 = x c b+cλ y c b+cλx +β Q = x c b+cλ Q+β Q. Similarly substituting the last equation of 7 and equations 17 16 and 8 in 9 we get that in the new variables f = b b+cλ y + c b+cλ x +λ x c b+cλx +β Q c = Q+λ x c b+cλ Q+β Q. By defining β 1 = c/b + cλ we obtain that the above f 1 and f satisfy the second Hamiltonian of the theorem now with β 1 0.

10 F. BRAUN J. LLIBRE AND A. C. MEREU We have yet to analyze possibility 18. The remaining part of the proof will be to show that if we assume this possibility we get a contradiction. We will treat this analyzing two cases: a = bλ and a+bλ 0. In the first case we consider new c 3 and c 4 not zero in order that 18 turn to 9 p 4 = c 4 y λx 4 q 3 = λp 3 +c 3 y λx 3. We take the change of coordinates xy xy λx. Then using 9 we write equations 10 11 1 and 13 in these new variables where we denote p i xy = p i xy λx and q i xy = q i xy λx i = 34: 30 p x +q λp y = 0 p 3x +p x q y q x p y +3c 3 y = 0 p 3x q λp y q λp x p 3y +3c 3 p x y = 0 4c 4 q λp x y +3c 3 p 3x y = 0. Integrating the fourth equation of 30 in y we obtain d 3 R such that p 3 = 4c 4 3c 3 q λp y +d 3 y 3. On the other hand defining p = a 1 x +a xy +a 3 y and integrating in y the first equation of 30 we obtain a 4 R such that q = a 4 x +a λ a 1 xy +a 3 λ a y. Then we substitute the above q and p 3 in the second and in the third equations of 30 obtaining two identically zero homogeneous polynomials of degrees and 3 respectively. We denote this by h = 0 and h 3 = 0 respectively. The coefficient of y 3 of h 3 is 8c 4 a 4 a 1 λ /3c 3. Since c 4 0 it follows that a 4 = a 1 λ. Then the coefficient of x of h turns to 4a 1 and hence a 1 = 0. Finally the coefficients of y and y 3 of h and h 3 respectively are 4a +3c 3 and 6a c 3 implying that a = c 3 = 0 which is a contradiction. In the second case we consider the change of variables xy ax + byy λx and write p i and q i i = 34 the maps p i and q i in these new variables i.e. p i xy = p i ax+byy λx q i xy = q i ax +byy λx. Then denoting the new variables by xy again it follows that p 4 = c 4 x 4 q 4 = λp 4 and q 3 = λp 3 +c 3 x 3 and equations 10 11 1 and 13 turn

ISOCHRONICITY FOR QUINTIC AND SEPTIC HAMILTONIAN SYSTEMS 11 respectively to 31 ap x λp y +bq x +q y = 0 a+bλ p 3x +p x q y q x p y +3bc 3 x = 0 a+bλ p 3x q λp y q λp x p 3y +4c 4 x 3 3c 3 p y x = 0 a+bλ 4c 4 q λp y x 3c 3 p 3y x = 0. The second and the fourth equations of 31 give respectively that 3 p 3x = p x q y q x p y 3bc 3 a+bλ x p 3y = 4c 4 3c 3 q λp y x. Substituting 3 in the third equation of 31 we obtain that p x q y p y q x + 4c 4 q 33 3c λp x x+ 3bc 3 3 a+bλ x q λp y 4c 4 x 3 +3c 3 p y x = 0. Then defining p = a 1 x + a xy + a 3 y the first equation of 31 will be a differential equation in q. Solving it for a homogeneous polynomial of degree now it is similar to Lemma but we apply the change of variables x x by instead and integrate in y it follows that there exists a 4 R such that q = a 4 x by aa 1 a λx byy aa +a 1 b a 3 +a bλy. Substituting the above p and q in 33 we obtain an identically zero homogeneous polynomial of degree 3. The coefficient of x 3 of this polynomial gives that 4c 4 3 aa 8a4 c 4 1 +a 4 b a λ + 9bc 3 = 0. c 3 a+bλ Now we substitute p and q in the equation p 3xy p 3yx = 0 given by 3 and obtain an identically zero homogeneous polynomial of degree 1. The coefficient of x of this polynomial gives that 16c 4 aa 1 +a 4 b a λ 3c 3 = 0. Combining the last two equations we obtain that c 4 = 0 a contradiction. We observe that the two hamiltonians that appear in Theorem 5 can not be transformed in each other using a linear change of coordinates. This is because applying the linear change xy ax+bycx+dy in the second Hamiltonian the only way to make the monomial x 8 disappear is to take

1 F. BRAUN J. LLIBRE AND A. C. MEREU a = 0. Then to make the monomial x 4 disappear we have to take c = 0. But then we no longer have a change of coordinates. We also observe that our results give formulas up to linear change of coordinates for all the polynomial maps f = f 1 f : R R such that detdf = 1 f00 = 00 anddegree of f is 3 or 4. Usingthese formulas it is very simple to see that such maps are injective. 3. Examples of polynomial Hamiltonian isochronous centers The following example shows that there exist non trivial polynomial Hamiltonian isochronous centers of degree 6k +1 for all k {1...}. Example 6. Let λ R and k {1...}. Let f = f 1 f : R R be defined by x+λy k x+λy k + 1+x+λy k f 1 = 1+x+λy k f y =. 1+x+λy k It follows that the Jacobian determinant of f is constant and equal to 1. Moreover taking H = f 1 +f / it is simple to see that H is the polynomial x+λy k +yx+λy k 1+x+λy k +y 1+x+λy k 3 which clearly has degree 6k+ if λ 0 and degree 8 if λ = 0. Thus system 1 with Hamiltonian H has an isochronous center of degree 6k + 1 at the origin if λ 0 degree 7 if λ = 0. Lemma 7. The isochronous center presented in Example 6 is non-trivial. Proof. Supposeon thecontrary that thecenter is trivial. Thenthereexists a polynomial map g = g g : R R with detdg = 1 and g00 = 00 such that g 1 +g = H. The map h = g T with Txy = x λyk y is also polynomial detdh = 1 h00 = 00 and H = h 1 +h = x +yx 1+x +y 1+x 3 is a polynomial of degree 8. Thus system 1 with Hamiltonian H has a trivial isochronous center at the origin. Since h is globally injective by the mentioned result of [7] or by Theorem 5 it follows from the mentioned result of [8] that this center is global. This is not possible because the level curve H = 1/ is not bounded it is formed by the curves y = 1/1+x and y = 1 x /1+x. Example 6 with λ = 0 has already appeared in []. The following example provide trivial isochronous centers for all even degrees.

ISOCHRONICITY FOR QUINTIC AND SEPTIC HAMILTONIAN SYSTEMS 13 Example 8. Let k {3...} λc...c k R with c k 0 and f = f 1 f : R R be defined by f 1 = x+c y + +c k y k f = y +λf 1. It is clear that f00 = 00 and detdf = 1. Thus system 1 with the Hamiltonian given by H = f1 +f / has a trivial polynomial Hamiltonian isochronous center of degree k 1 at the origin. Since f is clearly injective this center is global. Acknowledgments The first author is partially supported by a FAPESP-BRAZIL grant 014/6149-3. The second author is partially supported by a MINECO grant MTM 013 40998 P an AGAUR grant number 014SGR 568 and FP7-PEOPLE-01-IRSES-316338 and 318999. The third author is partially supported by a FAPESP-BRAZIL grant 01/18780-0 and a CNPq grant 449655/014-8. The three authors are also supported by a CAPES CSF PVE grant 88881.030454/ 013-01 from the program CSF-PVE References [1] K. Baba and Y. Nakai A generalization of Magnu s Theorem Osaka J. Math. 14 1977 403 409. [] A. Cima F. Mañosas and J. Villadelprat Isochronicity for Several Classes of Hamiltonian Systems J. Differential Equations 157 1999 373 413. [3] A. van den Essen Polynomial automorphisms and the Jacobian conjecture Progress in Mathematics 190. Birkhäuser Verlag Basel 000. [4] X. Jarque and J. Villadelprat Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four J. Differential Equations 180 00 334 373. [5] J. Llibre and V. G. Romanovski Isochronicity and linearizability of planar polynomial Hamiltonian systems J. Differential Equations 59 015 1649 166. [6] F. Mañosas and J. Villadelprat Area-Preserving Normalizations for centers of Planar Hamiltonian Systems J. Differential Equations 179 00 65 646. [7] T. T. Moh On the Jacobian conjecture and the configurations of roots J. Reine Angew. Math. 340 1983 140 1. [8] M. Sabatini A connection between isochronous Hamiltonian centers and the Jacobian Conjecture Nonlinear Anal. 34 1998 89 838. Department of Mathematics. UFSCar 13565-905 São Carlos SP Brazil E-mail address: franciscobraun@dm.ufscar.br Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra Barcelona Catalonia Spain E-mail address: jllibre@mat.uab.cat Department of Physics Chemistry and Mathematics. UFSCar. 1805-780 Sorocaba SP Brazil E-mail address: anamereu@ufscar.br