This is a preprint of: “The improved Euler-Jacobi formula and the planar cubic polynomial vector fields in R2 ”, Jaume Llibre, Clàudia Valls, Dyn. Syst., vol. 37(3), 403–443, 2022. DOI: [10.1080/14689367.2022.2066508] THE IMPROVED EULER-JACOBI FORMULA AND THE PLANAR CUBIC POLYNOMIAL VECTOR FIELDS IN R2 JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. The new Euler-Jacobi formula for points with multiplicity two provides an algebraic relation between the singular points of a poly- nomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topo- logical indices for the planar cubic polynomial differential systems when these systems have eight finite singular points, being one of them with multiplicity two. The case with nine finite singular points has already been solved using the classical Euler-Jacobi formula. 1. Introduction and statement of the main results Consider the planar polynomial differential system (1) ẋ = P (x, y), ẏ = Q(x, y), in R2 where P (x, y) and Q(x, y) are real polynomials of degree n and m respectively. The vector field associated to the planar polynomial differential system (1) will be denoted by X. When n = m = 2 and system (1) has four singular points the possible configurations of the (topological) indices of these points where characterized by Berlinskii in [2] in 1960. Later on in 1999 Gasull and Torregrosa [7] also for systems (1) with n = m = 2 but with three singular points one with multiplicity two extended the Berlinskii theorem to these systems. More recently, the authors in [9] with n = m = 3 (i.e. for cubic polynomial differential systems) classify all the possible configurations of the indices of these systems having nine singular points. The objective of this paper is to classify all the configurations of indices of systems (1) with n = m = 3 having eight singular points one of them with multiplicity two. When system (1) has nm finite singular points the classical Euler-Jacobi formula (see [1] for its proof) gives an algebraic relation between the indices of these finite singular points. When system (1) has points with multiplicity two, the classical Euler-Jacobi formula is no longer valid. However in [7] the 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. Euler-Jacobi formula, singular points, topological index, polyno- mial differential systems. 1 2 J. LLIBRE AND C. VALLS authors (Gasull and Torregrosa) provided a generalization of the classical Euler-Jacobi formula when the system has points with multiplicity two. Using this new formula and the index theory we obtain the distribution of the singular points of cubic polynomial differential systems when this system has eight finite singularities, being one of them with multiplicity two. We denote by AX = A the set of points p ∈ R2 such that X(p) = 0. Given a finite subset B of R2 , we denote by B̂, ∂ B̂ and #B its convex hull, the boundary of the convex hull, and its cardinal, respectively.  Set A0 = A and Ai = A ∩ ∂ A \ A0 ∪ \ · · · ∪ Ai−1 for i ≥ 1. Note that there exists q ∈ N such that Aq 6= ∅ and Aq+1 = ∅. We say that A has the configuration (K0 ; K1 ; K2 ; . . . ; Kq ) if Ki = #(Ai ∩ ∂ Âi ). We say that A has the configuration (K0 ; K1 ; K2 ; . . . ; Kr ; ∗) if we do not specify for the values of Ki for i between r + 1 and q. We also say that the singular points of X belonging to Ai ∩ ∂ Âi are on the i-th level. We want to be more precise and study also the indices of the singular points of X. Then we substitute each Ki by the sign of the indices of the points of Ai ∩ ∂ Âi , i.e. instead of Ki in the configuration we write the string (si1 , si2 , . . . , siKi ) where sji ∈ {+, −, 0}. When Ai ∩ ∂ Âi is a polygon, the starting si1 will be the point with multiplicity two (denoted by p0 ) if such point is in the i-th level and the signs si2 , . . . , siKi are the signs of the list of positive or negative indices that follow point with multiplicity two in counterclockwise or clockwise sense according with the largest list of points between the two lists closest to the point with multiplicity two. In case that both closest lists have the same length, we choose the one with the second largest closest list, and so one. In fact when there are ` equal consecutive signs, for instance if they are +, then instead of + · · · + `-times we shall write `+. In the case that the i-th level does not contain the point with multiplicity two, then si1 is the length of the largest list of positive or negative indices of the singular point in the i-th level. The numbers si2 , . . . , siK1 are chosen following the previous criteria changing the point with multiplicity two by si1 . When Ai ∩ ∂ Âi is a segment, which does not contain the point with multiplicity two, the signs of the strings are ordered starting at one of the endpoints. Then we start for the endpoint having the larger list of signs independently if this list is formed by plus or minus signs. In case that the length of the list of signs of both endpoints are equal, then we choose to start with the endpoint whose second list is larger, and so on. If Ai ∩ ∂ Âi contains the point with multiplicity two we identify all the list of signs of this segment cyclically, i.e. after one endpoint it follows the other endpoint. Then the starting sign in the list is the sign 0 of the point of multiplicity two, and after it we choose the largest list closest to p0 . In THE IMPROVED EULER-JACOBI FORMULA 3 case that the two lists of signs separated by p0 have the same length, then we choose to start with the list near p0 whose second list is larger, and so on. With this notation we can state the main result of the paper. We denote by iX (a) the index of a singular point a ∈ A of a planar cubic polynomial vector field X. It was proved in [6] and [8] that in the case of cubic polynomial differential systems the absolute value of the sum of the indices of the points is either 1 or 3. In the next theorem, which is our first main theorem we characterize all the possible configurations for cubic polynomial differential systems when the absolute value of the sum of the indices is three. Theorem 1. For planar cubic polynomial differentialP systems having 8 sin- gular points with one of multiplicity two, and | a∈A iX (a)| = 3, the only possible configurations for the eight topological indices of their singularities are (5;3) (5+; 0, 2−), (0, 4+; 2−, +), (5−; 0, 2+), (0, 4−; 2+, −). (4;4) (4+; 0, 2−, +), (4+; 0, 2−, +), (0, 3+; +, −, +, −), (0, 3+; 2+, 2−), (4−; 0, 2+, −), (4−; 0, +, −, +), (0, 3−; +, −, +, −), (0, 3−; 2+, 2−). (4;3;1) (4+; +, 2−; 0), (4+; 0, 2−; +), (0, 3+; 2−, +; +), (0, 3+; 2+, −; −), (4−; 2+, −; 0), (4−; 0, 2+; −), (0, 3−; 2+, −; +), (0, 3−; 2−, +; +). (3;5) (3+; 0, +, −, +, −), (0, 2+; 2+, −, +, −), (3−; 0, −, +, −, +), (0, 2−; 2−, +, −, +). (3;4;1) (3+; 2+, 2−; 0), (3+; +, −, +, −; 0), (0, 2+; +, −, +, −; +), (3−; 2+, 2−; 0), (3−; −, +, −, +; 0), (0, 2−; −, +, −, +; −). (3;3;2) (3+; 2+, −; 0, −), (3+; 0, −, +; +, −), (0, 2+; 2+, −; +, −), (3−; 2−, +; 0, +), (3−; 0, +, −; +, −), (0, 2−; 2−, +; +, −). and there exist examples of such cubic polynomial differential systems with these configurations. The proof of Theorem 1 is given in section 3. To complete the P cubic case it is necessary to characterize all possible configurations when | a∈A iX (a)| = 1 which is done in the following theorem. Theorem 2. For planar cubic polynomial differentialP systems having 8 sin- gular points with one of multiplicity two, and | a∈A iX (a)| = 1, the only possible configurations for the eight topological indices of their singularities are (8) (0, +, −, +, −, +, −, +), (0, 2+, −, +, −, +, −), (0, −, +, −, +, −, +, −) and (0, 2−, +, −, +, −, +). (7;1) (0, +, −, +, −, +, −; +), (0, 2+, −, +, −, +; −), (0, 3+, −, +, −; −), (2+, −, +, −, +, −; 0), (0, −, +, −, +, −, +; −), (0, 2+, −, +, −, +; −), (0, 3−, +, −, +; +) and (2−, +, −, +, −, +; 0). 4 J. LLIBRE AND C. VALLS (6;2) (0, +, −, +, −, +; +, −), (0, −, 2+, −, +; +, −), (0, 2+, −, +, −; +, −), (0, 2+, −, 2+; 2−), (2+, −, 2+, −; 0, −), (3+, −, +, −; 0, −), (+, −, +, −, +, −; 0, +), (0, −, +, −, +, −; +, −), (0, +, 2−, +, −; +, −), (0, 2−, +, −, +; +, −), (0, 2−, +, 2−; 2+), (2−, +, 2−, +; 0, +), (3−, +, −, +; 0, +) and (+, −, +, −, +, −; 0, −). (5;3) (0, −, 2+, −; 2+, −), (0, +, −, +, −; 2+, −), (0, 2+, −, +; 2−, +), (0, 3+, −; 2−, +), (2+, −, +, −; 0, +, −), (4+, −; 0, 2−), (0, +, 2−, +; 2−, +), (0, −, +, −, +; 2−, +), (0, 2−, +, −; 2+, −), (0, 3−, +; 2+, −), (2−, +, −, +; 0, −, +) and (4−, +; 0, 2+). (4;4) (+, −, +, −; 0, +, −, +, −), (+, −, +, −; 0, 2+, −), (0, −, +, −; 3+, −), (0, +, −, +; +, −, +, −), (0, +, −, +; 2+, 2−), (0, 2+, −; +, −, +, −), (0, 2+, −; 2+, 2−), (0, 3+; 3−, +), (3+, −; 0, −, +, −), (3+, −; 0, 2−, +), (4+; 0, 3−), (+, −, +, −; 0, −, +, −, +), (+, −, +, −; 0, 2−, +), (0, +, −, +; 3−, +), (0, −, +, −; +, −, +, −), (0, −, +, −; 2+, 2−), (0, 2−, +; −, +, −, +), (0, 2−, +; 2+, 2−), (0, 3−; 3+, −), (3−, +; 0, +, −, +), (3−, +; 0, 2+, −) and (4−; 0, 3+). (4;3;1) (0, +, −, +; 2−, +; +), (+, −, +, −; 0, +, −; +), (+, −, +, −; 0, 2+; −), (+, −, +, −; 2+, −; 0), (0, −, +, −; 2+, −; +), (0, −, +, −; 3+; −), (0, 2+, −; 2+, −; −), (0, 3+; 3−; +), (3+, −; 2−, +; 0), (3+, −; 0, +, −; −), (3+, −; 0, 2−; +), (4+; 3−; 0), (4+; 0, 2−; −), (0, 2+, −; 2−, +; +), (0, −, +, −; 2+, −; +), (0, −, +, −; 2+, −; −), (−, +, −, +; 0, −, +; −), (−, +, −, +; 0, 2−; +), (−, +, −, +; 2−, +; 0), (0, −, +, −; 2−, +; −), (0, +, −, +; 3−; +), (0, 2−, +; 2−, +; +), (0, 3−; 3+; −), (3−, +; 2+, −; 0), (3−, +; 0, −, +; +), (3−, +; 0, 2+; −), (4−; 3+; 0), (4−; 0, 2+; +), (0, 2−, +; 2+, −; −) and (0, +, −, +; 2−, +; −). (3;5) (0, +, −; 2+, −, +, −), (0, 2+; 2−, +, −, +), (2+, −; 0, −, 2+, −), (2+, −; 0, +, −, +, −), (3+; 0, 2−, +, −), (0, +, −; 2−, +, −, +), (0, 2−; 2+, −, +, −), (2−, +; 0, +, 2−, +), (2−, +; 0, −, +, −, +) and (3−; 0, 2+, −, +). (3;4;1) (0, +, −; +, −, +, −; +), (0, +, −; 3+, −; −), (0, 2+; 3−, +; −), (2+, −; 0, 2−, +; +), (2+, −; 0, 2+, −; −), (2+, −; 2+, 2−; 0), (3+; 0, 2−, +; −), (3+; 0, 3−; +), (3+; 3−, +; 0), (0, −, +; −, +, −, +; −), (0, −, +; 3−, +; +), (0, 2−; 3+, −; +), (2−, +; 0, 2+, −; −), (2−, +; 0, 2−, +; +), (2−, +; 2−, 2+; 0), (3−; 0, 2+, −; +), (3−; 0, 3+; −), and (3−; 3+, −; 0), (3;3;2) (0, +, −; 2+, −; +, −), (0, 2+; 2−, +; +, −), (0, 2+; 3−; 2+), (2+, −; 2−, +; 0, +), (2+, −; 0, +, −; +, −), (2+, −; 2+, −; 0, −), (3+; 0, 2+; +, −), (2+, −; 0, 2−; 2+), (3+; 3−; 0, +) (0, −, +; 2−, +; +, −), (0, 2−; 2+, −; −, +), (0, 2−; 3+; 2−), (2−, +; 2+, −; 0, −), (2−, +; 0, −, +; −, +), (2−, +; 2−, +; 0, +), (3−; 0, 2−; +, −), (2−, +; 0, 2+; 2−) and (3−; 3+; 0, −). and there exist examples of such cubic polynomial differential systems with these configurations. THE IMPROVED EULER-JACOBI FORMULA 5 The proof of Theorem 2 is given in section 4. As mentioned above with Theorems 1 and 2 we provide the classification of the configurations of the singular points and their topological indices for all planar polynomial dif- ferential systems with eight finite singular points being one (that we will translate it at the origin) double. 2. Preliminaries First of all we observe that if a configuration exists for a cubic polynomial vector field X with #AX = 8, being one of them double then it is possible to construct the same configuration but interchanging points with index +1 with points with index −1. For doing that it is enough to take Y = (−P, Q) instead P of X = (P, Q). So we can restrict ourselves to the cases in which a∈A X (a) ≥ 0. Moreover it was proved in [6] and [8] that in this case i the sum of the absolute value of the indices of the points is either 3 or 1. Since there is only one point with multiplicity two we must have five points with positive index and two points with negative index when the sum of the indices of the finite singular points is three, and four points with positive index and three points with negative index when the sum of the indices of the finite singular points is one. It follows from index theory (see for instance [3]) that if two points collide then the indices add. Since the indices of the semi-hyperbolic points can only be −1, 1, 0 and the indices of the hyperbolic points when there are nine finite singular points can only be −1, 1 it follows that only points with indices −1 and +1 can collide. In short, the only possible configurations in Theorem 1 can be from colliding two hyperbolic points with index +1 and −1 in a configuration P of a planar cubic polynomial differential system such that #A = 9 and | a∈A iX (a)| = 3. In [9] the authors obtained such configurations in the following theorem. Theorem 3. For P planar cubic polynomial differential systems having 9 sin- gular points and | a∈A iX (a)| = 3 the only possible configurations for the nine topological indices of their singularities are (5;3;1) with (5+; 3−; +), (5−; 3+; −); (4;5) with (4+; +, 2−, +, −), (4−; 2+, −, +, −); (4;4;1) with (4+; +, 3−; +), (4−; 3+, −; −); (4;3;2) with (4+; 3−; 2+), (4+; +, 2−; +, −), (4−; 3+; 2−), (4−; 2+, −; +, −); (3;6) with (3+; +, −, +, −, +, −), (3−; +, −, +, −, +, −); (3;5;1) with (3+; +, 2−, +, −; +), (3−; 2+, −, +, −; −); (3;4;2) with (3+; 2+, 2−; +, −), (3−; 2+, 2−; +, −); (3;3;3) with (3+; 2+, −; +, 2−), (3−; +, 2−; 2+, −); and there exist examples of such cubic polynomial differential systems with these configurations. 6 J. LLIBRE AND C. VALLS Lemma 4. For planar cubic polynomial differential P systems with 8 singular points being one with multiplicity two, and | a∈A iX (a)| = 3, the only pos- sible configurations for the eight topological indices of their singularities are of the form (K+; ∗) where K ≤ 5 or (0, L+; ∗) where L ≤ 4. Proof. It follows directly from Theorem 3 and the fact that the point with multiplicity two can only formed by colliding two hyperbolic points in a configuration P of planar cubic polynomial differential systems such that #A = 9 and | a∈A iX (a)| = 3.  It follows that a singular point p of system (1) is simple if and only if the determinant of the Jacobian matrix of P and Q at p is different from zero, i.e.   ∂P ∂Q ∂P ∂Q J(P, Q)(p) := J(p) = − 6= 0. ∂x ∂y ∂y ∂x p A singular point p of system (1) is double if and only if J(p) = 0 and I(P, Q)(p) := I(p) 6= 0 where     ∂P 2 ∂P ∂ 2 Q ∂Q ∂ 2 P I(P, Q)(p) := I(p) = − ∂y ∂x ∂x∂x2 ∂x ∂x2   ∂P ∂P ∂P ∂ 2 Q ∂Q ∂ 2 P −2 − ∂x ∂y ∂x ∂x∂y ∂x ∂x∂y  2   ∂P ∂P ∂ Q ∂Q ∂ 2 P 2 + − . ∂x ∂x ∂y 2 ∂x ∂y 2 p For a proof see Lemma 2.2 of [7]. Moreover it is well-known that for planar polynomial differential systems (1), a simple point p has index 1 (if J(p) > 0) or −1 (if J(p) < 0) (see for instance [10]) and that points with multiplicity two of our system (1) has index zero. In this paper we consider the case in which one of the 8 singular points, that we will denote by p0 , has multiplicity two. We will use the new Euler- Jacobi formula for points with multiplicity two proved by Gasull and Torre- grosa in [7] which can be stated as follows. To state it we need the following notation. We write P (x, y) = P10 x + P01 y + P20 x2 + P11 xy + P02 y 2 + . . . , Q(x, y) = Q10 x + Q01 y + Q20 x2 + Q11 xy + Q02 y 2 + . . . and given a polynomial R we also write it as R(x, y) = R00 + R10 x + R01 y + R20 x2 + R11 xy + R02 y 2 + . . . The next result is proved in Theorem 3.2 of [7] for two real polynomials of degrees n and m. We state it here for the case in which n = m = 3 and when the system has eight finite singular points one with multiplicity two. THE IMPROVED EULER-JACOBI FORMULA 7 Theorem 5. Consider a cubic system of two real polynomials in the vari- ables x and y. If the set of zeroes of that system (that we denote by A) contains exactly eight elements (seven being simple and one with multiplic- ity two p0 that without loss of generality we can assume it is at the origin), then for any polynomial R of degree less than or equal to 3 we have X R(a) (2) + S(0) = 0 J(a) a∈AS where AS denotes the set of simple zeroes of the system and S(0) is equal to 4P10 R00 N 2P10 (P10 R01 − P01 R10 ) S(0) = 2 + , I(0) I(0) where 3 2 N = P10 (Q10 P03 − P10 Q03 ) − P10 P01 (Q10 P12 − P10 Q12 ) 2 3 + P10 P01 (Q10 P21 − P10 Q21 ) − P01 (Q10 P30 − P10 Q30 ) 3 2 + P10 (Q11 P02 − P11 Q02 ) − 2P10 P01 (Q20 P02 − P20 Q02 ) 2 + P10 P01 (Q20 P11 − P20 Q11 ). In the proof of Theorems 1 and 2 we will denote by Lij the straight line through the points pi and pj where i, j ∈ {1, . . . , 8} and also L2ij = Lij Lij . Moreover during all the paper we will consider straight lines with only two singular points because if there are straight-lines with three singular points the arguments become even easier. Furthermore, we note that when proving that some configurations are not possible, we will take p0 = (0, 0) and a conic (generally formed by two straight lines) passing through p0 . Doing so, we have that R00 = R10 = R01 = 0, and consequently S(0) = 0 in formula (2). LemmaP6. For planar cubic polynomial differential systems such that #AX = 8, and | a∈A iX (a)| = 1, there are no configurations of the form (K∗, 2−, ∗; ∗), that is, there are no configurations with two consecutive points with negative index in the 0-level. Proof. Assume that there are two consecutive points in the 0-level with negative index and denote them by p1 , p2 . Since the absolute value of the sum of the indices is one and we can assume that the sum is positive, we have an additional point with negative index, that we denote by p3 . Hence, applying the Euler-Jacobi formula (2) with R = L20,p3 Lp1 ,p2 we reach a con- tradiction.  3. Proof of Theorem 1 It follows from Lemma 4 that the possible configurations are: (5+; 3), (0, 4+; 3), (4+; 4), (0, 3+; 4), (4+; 3; 1), (0, 3+; 4; 1), (3+; 5), (0, 2+; 5), (3+; 4; 1), 8 J. LLIBRE AND C. VALLS (0, 2+; 4; 1), (3+; 3; 2) and (0, 2+; 3; 2). We will study each of them sepa- rately. Configuration (5+; 3) Since there are five points with positive index and two points with negative index the unique possible configuration is (5+; 0, 2−). The cubic system (1) with P (x, y) = −6y + 5y 2 + 2x2 y − y 3 , 1 Q(x, y) = (48 + 32x − 40y − 12x2 − 26xy + 8y 2 − 8x3 − x2 y + 5xy 2 ), 5 has the singular points (−1., 4.), (0, 3), (−2, 0), (2, 0), (1, 4), (−1, 1), (−3/2, 0), (0, 2), (0, 2) in the configuration (5+; 0, 2−). Configuration (0, 4+; 3) Since there are five points with positive index and two points with negative index the unique possible configuration is (0, 4+; 2−, +). The cubic system (1) with P (x, y) = 6y − 5y 2 − 2x2 y + y 3 , 1 Q(x, y) = − (48x + 3y − y 2 − 46xy − 12x3 − 2x2 y + 10xy 2 ), 10 has the singular points (0, 3), (0, 3), (−2, 0), (2, 0), (1, 4), (−1, 1), (0.428571, 1.71429), (1, 1), (0, 0) in the configuration (0, 4+; 2−, +). Configuration (4+; 4) Since there are five points with positive index and two points with negative index the unique possible configurations of the form (4+; 4) are (4+; 0, −, +, −) and (4+; 0, 2−, +). The cubic system (1) with P (x, y) = 1.4523 + 1.23689x + 3.08149y + 3.84729x2 + 1.05231xy − 3.52134y 2 + 0.168176x3 + 0.910772x2 y − y 3 , Q(x, y) = 1.2082 + 11.867x + 14.1512y + 2.98455x2 + 7.00776xy + 3.81719y 2 + 0.0233874x3 + 0.827415x2 y + xy 2 , has the singular points (−3.9, −3.6), (−3.92, −3.5), (−3.84.., −3.5..), (−3.84.., −3.5..), (16.24.., −13.16..), (3.99.., −3.74..), (−2.99.., 2.82..), (2.00., −1.73..), (0.86.., −0.80..) in the configuration (4+; 0, −, +, −). THE IMPROVED EULER-JACOBI FORMULA 9 The cubic system (1) with P (x, y) = −2.14508 − 2.27088x − 0.61618y − 1.93249x2 + 0.32733xy + 2.46863y 2 + 0.26016x3 + 1.06646x2 y − y 3 , Q(x, y) = −5.23367 − 4.92581x − 0.81582y − 3.21993x2 − 0.9972xy + 2.79274y 2 − 0.78125x3 − 0.24175x2 y + xy 2 , has the singular points (−4, 3), (3, −3), (−3.29.., 2.66..), (−3, 2.82..), (4, 5), (−0.95.., 1.25..), (−0.95.., 1.25..), (−3, −2.82..), (−2, −1.73..) in the configuration (4+; 0, 2−, +). Configuration (0, 3+; 4) Since there are five points with positive index and two points with negative index the unique possible configurations are (0, 3+; +, −, +, −) and (0, 3+; 2+, 2−). The cubic system (1) with P (x, y) = −2.14507 − 2.27088x − 0.61617y − 1.93248x2 + 0.32733xy + 2.46863y 2 + 0.26016x3 + 1.06646x2 y − y 3 , Q(x, y) = −5.23368 − 4.92581x − 0.81582y − 3.21993x2 − 0.99719xy + 2.79274y 2 − 0.78125x3 − 0.24175x2 y + xy 2 , has the singular points (−4, 3), (3, −3), (−3.29.., 2.66..), (−3, 2.82..), (4, 5), (−0.95.., 1.25..), (−0.95.., 1.25..), (−3, −2.82), (−2, −1.73) in the configuration (0, 3+; +, −, +, −). The cubic system (1) with P (x, y) = 0.00003 − 0.00002x + 0.19126y − 0.00002x2 + 0.46652xy + 0.11736y 2 + 0.98626x2 y − y 3 , Q(x, y) = 0.00001 − 1.09067y + 0.16407xy + 2.18129y 2 + 0.43317x2 y + xy 2 , has the singular points (−3.00.., 2.82..), (−5.05.., 0.00..), (4.00.., 0.00..), (4.00.., 0.00..), (−0.73.., 0.67..), (−1.50.., 0), (0.99.., 0), (0, 0.49..), (−2.00.., −1.73..) in the configuration (0, 3+; 2+, 2−). Configuration (4+; 3; 1) Since there are five points with positive index and two points with negative index the unique possible configurations are (4+; 2−, +; 0), (4+; 0, 2−; +) and (4+; 0, +, −; −). 10 J. LLIBRE AND C. VALLS We will show that the configuration (4+; 0, +, −; −) is not possible. De- note by p1 , p2 , p3 , p4 the four points in the 0-level (with positive index), by p0 , p5 , p6 the points in the 1st-level (being p0 the point with multiplicity two, p5 the point with positive index and p6 the singular point with neg- ative index) and by p7 the singular point in the 2nd level. There exists k0 , k1 ∈ {1, 2, 3, 4} so that Lk0 ,6 leaves pk1 on one side and the other five points in the other side. We denote by pk2 and pk3 the two remaining points in the 0-level. Note that we can consider, without loss of generality, that L0,k2 leaves k3 on one side and the other five points in the other side. Apply- ing the Euler-Jacobi formula to S = L0,5 L0,k2 Lk0 ,6 we reach a contradiction. The cubic system (1) with P (x, y) = 0.01043 + 0.08997x − 0.00631y − 0.0381x2 + 0.29991xy + 0.16308y 2 − 0.0623x3 − 0.69456x2 y + y 3 , Q(x, y) = 0.03992 − 0.38324x − 0.55026y + 0.0967x2 − 1.57747xy − 2.27773y 2 + 0.24662x3 − 0.41836x2 y − xy 2 , has the singular points (−3., −2.82..), (−1.5, 0), (1, 0), (−2, 1.73..), (4, −3), (0, −0.3), (0, −0.3), (0.44.., −0.41..), (−0.1, 0.12) in the configuration (4+; 2−, +; 0). The cubic system (1) with P (x, y) = −0.94486 − 1.31372x − 1.34391y − 2.35727x2 + 0.14990xy + 2.92903y 2 + 0.19308x3 + 1.08818x2 y − y 3 , Q(x, y) = −7.09812 − 5.92471x + 0.59531y − 1.43097x2 − 0.98248xy + 1.45502y 2 − 0.53935x3 − 0.39364x2 y + xy 2 , has the singular points (−4, 3), (−3, 2.82..), , (−3, 2.82..), (3, −2.82..), (3, 5), (0.02.., 2.03..), (−0.95.., 1.25..), (−3, −2.82..), (−2, −1.73..) in the configuration (4+; 0, 2−; +). Configuration (0, 3+; 3; 1) Since there are five points with positive index and two points with negative index the unique possible configurations are (0, 3+; 2−, +; +) and (0, 3+; 2+, −; −). The cubic system (1) with P (x, y) = 8.01755 + 13.6686x + 13.329y + 8.63431x2 + 18.4872xy + 6.48002y 2 + 3.14207x3 + 4.69624x2 y − y 3 , Q(x, y) = −1.58427 − 1.41114x + 1.3319y − 0.934538x2 − 0.607739xy + 2.32775y 2 − 0.692771x3 − 0.130663x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 11 has the singular points (−4, −4), (−4, 5), (−4, 5), (−13.50.., 11.16..), (−1.5, −0.5), (0.57.., −1.16..), (−1, 0.2), (2, −1.73..), (4, −3) in the configuration (0, 3+; 2−, +; +). The cubic system (1) with P (x, y) = 13.31010 + 6.56921x + 10.49492y + 0.55371x2 + 6.40611xy + 4.24196y 2 + 0.94893x3 + 1.993x2 y − y 3 , Q(x, y) = −1.51411 + 0.83433x + 7.4396y + 1.98657x2 + 0.93647xy + 1.48236y 2 − 0.36592x3 − 0.0735x2 y + xy 2 , has the singular points (−119.62.., 69.71..), (−4, −4), (45.73.., −24.47..), (−2.52.., −2.17..), (−4, 5), (−4, 5), (−1.5, −0.5), (1.85, −2), (2, −1.73..) in the configuration (0, 3+; 2+, −; −). Configuration (3+; 5). Since there are five points with positive index and two points with negative index the unique possible configurations are (3+; 0, +, −, +, −), (3+; 0, +, 2−, +) and (3+; 0, 2+, 2−). We show that the configurations (3+; 0, +, 2−, +) and (3+; 0, 2+, 2−) are not possible. For the configuration (3+; 0, +, 2−, +) denote by p1 , p2 , p3 the points in the 0-level and by p4 , p7 and p5 , p6 the points in the 1st level with positive and negative indices, respectively. The curve C = L0,5 L0,6 has different signs on p7 and on p4 . Moreover there exits k0 ∈ {1, 2, 3} so that C(pk0 ) has either the same sign than C(p4 ), or the same sign than C(p7 ). Applying formula (2) to Lk1 ,k2 C (note that here R0,0 = R1,0 = R0,1 = 0 and so S(0) = 0) with k1 , k2 ∈ {1, 2, 3} and ki 6= k0 for i = 1, 2 we reach to a contradiction. Hence this case is not possible. For the configuration (3+; 0, 2+; 2−) denote by p1 , p2 , p3 the points in the 0-level and by p4 , p5 the points in the 1st level with negative index. Note that there exists k0 ∈ {1, 2, 3} so that the curve L0,4 L0,5 evaluated on pk0 has the same sign as C evaluated on two positive indices of the 1st level. Applying formula (2) with C = L0,4 L0,5 Lk1 ,k2 where k1 , k2 ∈ {1, 2, 3} with k1 6= k0 and k2 6= k0 we reach to a contradiction. In short the unique possible configuration is (3+; 0, +, −, +, −). The cubic system (1) with P (x, y) = 0.10892 − 0.06352x − 1.52889y + −0.06353x2 − 2.2135xy + 0.09531y 2 + 0.01816x3 − 1.45684x2 y + y 3 , Q(x, y) = −1.43214 + 0.83542x + 1.82603y + x0.83542x2 + 0.31892xy + 1.74687y 2 − 0.23869x3 − 0.09658x2 y + xy 2 , 12 J. LLIBRE AND C. VALLS has the singular points (−3.00.., 2.83..), (4.00.., 0.00..), (0.15.., −1.45..), (−0.73.., 0.67..), (−0.73.., 0.67..), (−1.5.., 0), (−2.99.., −2.82..), (0.99.., 0), (−2.00.., −1.73..) in the configuration (3+; 0, +, −, +, −). Configuration (0, 2+; 5). Since there are five points with positive index and two points with negative index the unique possible configurations are (0, 2+; 3+, 2−) and (0, 2+; 2+, −, +, −). We will show that the configuration (0, 2+; 3+, 2−) is not possible. Indeed, denote by p1 , p2 the points in the 0 level and by p3 , p4 , p5 the points in the 1st level with positive index and by p6 , p7 the points in the 1st level with negative index. Applying formula (2) to R = L0,1 L0,2 L6,7 we reach to a contradiction. In short only the configuration (0, 2+; 2+, −, +, −) is possible. The cubic system (1) with P (x, y) = 0.24337 + 0.04054x − 0.28334y − 0.06592x2 + 0.51321xy + 0.09318y 2 + 0.01014x3 + 1.09144x2 y − y 3 , Q(x, y) = −1.69782 − 0.2829x + 2.22066y + 0.45983x2 − 0.1617xy + 2.35005y 2 − 0.07074x3 − 0.300636x2 y + xy 2 , has the singular points (−3.00.., 2.82..), (4.00.., 0.00..), (4.00.., 0.00..), (0.99.., −1.00..), (−1.5.., 0), (−0.49.., 0.46..), (−2.99.., −2.82..), (0, 0.499..), (−2.00.., −1.73..) in the configuration (0, 2+; 2+, −, +, −). Configuration (3+; 4; 1). Since there are five points with positive in- dex and two points with negative index the unique possible configurations are (3+; 2+, 2−; 0), (3+; +, −, +, −; 0), (3+; 0, 2−, +; +), (3+; 0, −, +, −; +), (3+; 0, +, −, +; −) and (3+; 0, 2+, −; −). We will show that the configura- tion (3+; 0, +, −, +; −) is not possible. Take the curve C = L0,4 L0,6 . Note that there exists k0 ∈ {1, 2, 3} so that it is the unique singularity con- tained in a connected component of R2 \ C. Applying formula (2) with C = L0,4 L0,5 Lk1 ,k2 where k1 , k2 ∈ {1, 2, 3} with k1 6= k0 and k2 6= k0 we reach a contradiction. The cubic system (1) with P (x, y) = 11.55589 − 4.10670x + 7.72417y + 1.08554x2 − 1.88945xy − 1.13607y 2 + 0.0411x3 + 0.42839x2 y − y 3 , Q(x, y) = 7.98585 + 5.44717x + 11.84708y + 0.7037x2 + 4.54680xy + 3.93911y 2 − 0.05877x3 + 0.40006x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 13 has the singular points (45.73.., −24.47..), (−4, 5), (1.85, −2), (2, −1.73..), (1, −1.5), (−4, −4), (−1.42.., −2.11..), (−2.52.., −2.17..), (−2.52.., −2.17..) in the configuration (3+; 0, −, +, −; +). The cubic system (1) with P (x, y) = 8.98459 + 6.6871x + 10.86426y + 2.87879x2 + 6.1108xy + 2.50325y 2 + 0.91190x3 + 2.00473x2 y − y 3 , Q(x, y) = 3.99511 + 1.65625x + 8.11454y − 0.114601x2 + 1.88344xy + 3.332y 2 − 0.27851x3 + 0.00545x2 y + xy 2 , has the singular points (45.73.., −24.47..), (14.69.., −8.08..), (−4, 5), (2, −1.73..), (1, −1.5), (1, −1.5), (−4, −4), (−1.5, −0.5), (−2.52.., −2.17..) in the configuration (3+; 2+, 2−; 0). The cubic system (1) with P (x, y) = 18.01525 − 0.56436x + 12.6791y + 1.95534x2 + 0.51759xy − 0.9431y 2 + 0.19838x3 + 0.7084x2 y − y 3 , Q(x, y) = 7.63906 + 5.25699x + 11.58106y + 0.657x2 + 4.41758xy + 3.92875y 2 − 0.06722x3 + 0.38503x2 y + xy 2 , has the singular points (45.73.., −24.47..), (−4, 5), (4, −3), (2, −1.73..), (1, −1.5), (−4, −4), (−2.52.., −2.17..), (−2.52.., −2.17..), (−2.96.., −2.14..) in the configuration (3+; +, −, +, −; 0). The cubic system (1) with P (x, y) = 12.53185 + 5.04636x + 15.68935y + 0.30004x2 + 2.48145xy + 2.9495y 2 + 0.03753x3 + 0.34625x2 y − y 3 , Q(x, y) = 7.74477 + 4.89906x + 11.2205y + 0.77834x2 + 4.32362xy + 3.69906y 2 − −0.04556x3 + 0.4297x2 y + xy 2 , has the singular points (45.73..., −24.47...), (−4, 5), (4, −3), (1.85, −2), (2, −1.73..), (2, −1.73..), (−1.5, −0.5), (−2.52.., −2.17..), (−3.51.., −2.36..) in the configuration (3+; 0, 2−, +; +). 14 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = 8.69530 + 2.74101x + 10.71702y + 0.56511x2 + 1.26514xy + 1.61948y 2 + 0.29941x3 + 0.85775x2 y − y 3 , Q(x, y) = 5.5855 + 4.1302x + 9.47293y + 0.9203x2 + 4.29167xy + 3.7649y 2 − 0.01426x3 + 0.49555x2 y + xy 2 , has the singular points (−4, −4), (45.73.., −24.47..), (−4, 5), (−1.5, −0.5), (0.8, −1.6), (0, 8, −1.6), (1.044.., −1.59..), (2, −1.73..), (4.2, −2.9) in the configuration (3+; 0, 2+, −; −). Configuration (0, 2+; 4; 1). Since there are five points with positive index and two points with negative index the unique possible configurations are (0, 2+; 2+, 2−; +), (0, 2+; +, −, +, −; +) and (0, 2+; 3+, −; −). We will show that the configuration (0, 2+; 2+, 2−; +) is not possible. Indeed, denote by p1 and p2 the points in the 0-level and by p3 , p4 the points in the 1st level with negative index. Applying formula (2) with C = L0,1 L0,2 L3,4 we reach a contradiction. The cubic system (1) with P (x, y) = 9.39252 − 15.4142x − 2.77515y + 3.34204x2 − 4.776xy − 4.97594y 2 + 0.07817x3 + 0.60294x2 y − y 3 , Q(x, y) = 4.63046 − 7.21976x − 0.67095y + 3.4358x2 + 1.63839xy − 0.27316y 2 + 0.04718x3 + 0.71861x2 y + xy 2 , has the singular points (−4, −4), (−4, −4), (−3.9, −4), (−4, 5), (1, −1.5), (45.73.., −24.47..), (1.85, −2), (2, −1.73..), (1.03.., −0.71..) in the configuration (0, 2+; +, −, +, −; +) The cubic system (1) with P (x, y) = 7.86131 − 22.585x − 10.2257y + 5.15825x2 − 5.92109xy − 7.34354y 2 + 0.21660x3 + 0.9303x2 y − y 3 , Q(x, y) = 2.50283 − 12.253x − 6.27431y + 4.96135x2 + 0.88691xy − 2.02498y 2 + 0.16245x3 + 0.99043x2 y + xy 2 , has the singular points (−4, −4), (−4, −4), (−4.1, 5.1), (−4, 5), (45.73.., −24.47..), (1.85, −3), (1.88.., −2.65..), (1, −1.85), (2, −1.73..) in the configuration (0, 2+; 3+, −; −). THE IMPROVED EULER-JACOBI FORMULA 15 Configuration (3+; 3; 2). Since there are five points with positive index and two points with negative index the unique possible configurations are (3+; 0, 2−; 2+), (3+; 0, 2+; 2−), (3+; 0, +, −; +, −), (3+; +, 2−; 0, +) and also (3+; 2+, −; 0, −). We will show that the configurations (3+; 0, 2−; 2+) is not possible. Indeed, we denote by p1 , p2 , p3 the points in the 0-level and by p4 , p5 the points in the 1st level with negative index. Note that there exists k0 ∈ {1, 2, 3} so that C = L0,4 L0,5 evaluated at pk0 has the same sign than C evaluated to the two positive points in the 2nd index. Applying formula (2) with C = L0,4 L0,5 Lk1 ,k2 where k1 , k2 ∈ {1, 2, 3} with k1 6= k0 and k2 6= k0 we have a contradiction. The cubic system (1) with P (x, y) = 7.80543 + 5.53466x + 11.72129y + 1.75211x2 + 3.947459xy + 2.83894y 2 + 0.4850x3 + 1.23003x2 y − y 3 , Q(x, y) = 3.70643 + 4.38387x + 6.9673y + 1.88746x2 + 5.093xy + 3.2218y 2 + 0.31337x3 + 1.135x2 y + xy 2 , has the singular points (−4, 5.20..), (4.92.., −3.88..), (4.00.., −2.99..), (3.82.., −2.91..), (1 − 85, −2), (1.85, −2), (1, −1.49..), (−1.96.., −0.00..), (−2.52.., −2.17..) in the configuration (3+; 0, 2+; 2−). The cubic system (1) with P (x, y) = 17.676 + 12.71329x + 23.36367y + 1.89862x2 + 5.68740xy + 3.79344y 2 + 0.00005x3 + 0.45244x2 y − y 3 , Q(x, y) = 10.45245 + 9.29009x + 14.92425y + 1.9876x2 + 6.28217xy + 3.87415y 2 − 0.01806x3 + 0.60356x2 y + xy 2 , has the singular points (−54.84.., −4.87..), (−4, 5.2), (4.92.., −3.88..) (4.00.., −2.99..), (4.00.., −2.99..), (1.85, −2), (1, −1.49..), (−1.96.., −0.00..), (−2.52.., −2.17..) in the configuration (3+; 0, +, −; +, −). The cubic system (1) with P (x, y) = 19.5475 + 16.6124x + 18.99y + 6.43719x2 + 10.846xy + 2.22562y 2 + 1.25946x3 + 2.69803x2 y − y 3 , Q(x, y) = 0.292006 − 5.41301x + 4.45102y − 2.412x2 − 1.63724xy + 2.91582y 2 − 0.598931x3 − 0.619738x2 y + xy 2 , 16 J. LLIBRE AND C. VALLS has the singular points (−4, −4), (45.73.., −24.47..), (−2.52.., −2.17..), (−4, 5), (−0.9.., −1.35..), (−0.9, −1.35), (−0.9, −1.35), (−0.9, −1.34), (4, −3) in the configuration (3+; +, 2−; 0, +). The cubic system (1) with P (x, y) = 6.23278 + 4.39091x + 9.86634y + 1.72876x2 + 3.67023xy + 2.68686y 2 + 0.56225x3 + 1.35392x2 y − y 3 , Q(x, y) = 3.14549 + 3.97591x + 6.30566y + 1.87914x2 + 4.99412xy + 3.16756y 2 + 0.34093x3 + 1.1792x2 y + xy 2 , has the singular points (−4, 5.20), (4.92.., −3.88), (4.00.., −2.99..), (1.85, −2), (1, −1.49..), (1, −1.49..), (−1.96.., −0.00..), (−1.36.., −0.31..), (−2.52.., −2.17..) in the configuration (3+; 2+, −; 0, −). Configuration (0, 2+; 3; 2). Since there are five points with positive index and two points with negative index the unique possible configurations are (0, 2+; 2−, +; 2+), (0, 2+; 3+; 2−) and (0, 2+; 2+, −; +, −). We show that the configurations (0, 2+; +, 2−; 2+) and (0, 2+; 3+; 2−) are not possible. For the configuration (0, 2+; 2−, +; 2+) denote by p1 , p2 the points in the 0 level and by p3 , p4 the two points with negative index in the 1st level. Applying formula (2) to R = L0,1 L0,2 L3,4 we reach a contradiction . For the configuration (0, 2+; 3+; 2−) we denote by p1 , p2 the points in the 0 level, by p3 , p4 , p5 the points in the 1st level and by p6 , p7 the points in the 2nd level. Note that there exists k0 ∈ {1, 2} and k1 ∈ {3, 4, 5} so that C = L0,k1 Lk2 ,k3 with k2 , k3 ∈ {3, 4, 5} such that k2 6= k1 and k3 6= k1 satisfies that C(pk0 ) has different sign than C(p6 ). Applying formula (2) with R = CL0,k4 with k4 ∈ {1, 2} and k4 6= k0 , we have a contradiction. In short only the configuration (0, 2+; 2+, −; +, −) is possible. The cubic system (1) with P (x, y) = 6.80256 − 22.2257x + 10.2134y − 4.99883x2 + 6.01111xy + 7.14855y 2 − 0.19868x3 − 0.89485x2 y + y 3 , Q(x, y) = 1.09836 − 11.7764x − 6.25806y + 4.74987x2 + 0.76750xy − 1.76632y 2 + 0.13868x3 + 0.9434x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 17 has the singular points (−4., −4), (−4, −4), (−4, 5), (−4.1, 5.1), (45.73.., −24.47..), (0.92.., −3.93..), (1, −1.85), (1.85, −2), (2, −1.73..) in the configuration (0, 2+; 2+, −; +, −). 4. Proof of Theorem 2 The possible configurations are (8), (7; 1), (6; 2), (5; 3), (4; 4), (4; 3; 1), (3; 5), (3; 4; 1) and (3; 3; 2). We will study each of them separately. Configuration (8) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, +, −, +, −, +, −, +) and (0, 2+, −, +, −, +, −). The cubic system (1) with P (x, y) = 2 − y + 2x2 − 2y 2 − x2 y + y 3 , Q(x, y) = 2.64643 − 0.41589x + 0.91936y − 2.46187x2 + 0.54440xy − 1.7054y 2 + 0.58412x3 − 0.22935x2 y + xy 2 , has the singular points (2, 2), (2, 2), (1, 2), (−0.58.., −0.80..), (−0.70.., −0.70..), (−0.86.., −0.5), (−0.97.., −0.22..), (0.58.., −0.80..), (0.86.., −0.5) in the configuration (0, +, −, +, −, +, −, +). The cubic system (1) with P (x, y) = 2 − 2x2 − y + x2 y − 2y 2 + y 3 , Q(x, y) = 2.61768 − 0.68657x + 1.04665y − 2.44229x2 + 0.30222xy − 1.56953y 2 + 0.78357x3 − 0.346x2 y + xy 2 , has the singular points (2, 2), (1, 2), (1, 2), (−0.58.., −0.80..), (−0.70.., −0.70..), (−0.14.., −0.98..), (−0.86.., −0.5), (0.42.., −0.90..), (0.86.., −0.5) in the configuration (0, 2+, −, +, −, +, −). Configuration (7; 1) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, +, −, +, −, +, −; +), (0, 2+, −, +, −, +; −), (0, 3+, −, +, −; −) and (2+, −, +, −, +, −; 0). 18 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = 128.006 + 265.684x − 73.0216y − 93.2801x2 + 260.74xy − 188.824y 2 − 180.419x3 + 125.935x2 y + y 3 , Q(x, y) = 151.198 + 312.596x − 83.878y − 110.076x2 + 307.755xy − 221.865y 2 − 211.959x3 + 146.972x2 y + xy 2 , has the singular points (2, 2), (1, 2), (−0.58.., −0.80..), (−0.70.., −0.70..), (−0.86.., −0.5), (−0.86.., −0.5), (−0.84.., −0.49..), (0.42.., −0.90..), (0.86.., −0.5) in the configuration (0, +, −, +, −, +, −; +). The cubic system (1) with P (x, y) =2 − y − 2x2 − 2y 2 + x2 y + y 3 , Q(x, y) = − 8.08705 − 1.52929x − 3.76842y + 4.96361x2 − 1.23536xy + 3.90598y 2 − 2.48180x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (1, 2), (−1.11.., 2), (−0.70.., −0.70..), (−0.86.., −0.5), (0.42.., −0.90..), (0.58.., −0.80..) in the configuration (0, 2+, −, +, −, +; −). The cubic system (1) with P (x, y) =4.64583 + 5.30182x − 1.50138y − 1.3072x2 + 0.45597xy − 6.5235y 2 − 6.98704x3 + 7.47839x2 y + y 3 , Q(x, y) =1.21683 + 2.34265x − 1.07425y − 0.99325x2 + 2.17057xy − 2.65934y 2 − 2.10737x3 + 1.08679x2 y + xy 2 , has the singular points (1.5, 2), (1.5, 2), (1, 1.5), (1.24.., 0.57..), (−0.86.., −0.5), (−0.70.., −0.5), (0.24.., −0.94), 0.42.., −0.90..), (0.58.., −0.80..) in the configuration (0, 3+, −, +, −; −) . The cubic system (1) with P (x, y) = − 5.45495 − 15.7188x + 3.26104y + 3.40044x2 − 15.4262xy + 9.05317y 2 + 10.6742x3 − 6.3916x2 y + y 3 , Q(x, y) = − 4.94356 − 16.6295x + 5.36845y + 3.03514x2 − 15.344xy + 9.64123y 2 + 11.61x3 − 7.84299x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 19 has the singular points (2, 2), (1, 2), (−0.58.., −0.80..), (−0.71.., −0.71..), (−0.70.., −0.70..), (−0.70.., −0.70..), (−0.86.., −0.5), (0.42.., −0.90..), (0.86.., −0.5) in the configuration (2+, −, +, −, +, −; 0). Configuration (6; 2) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, 3+, −, +; 2−), (0, 4+, −; 2−), (0, −, 2+, −, +; +, −), (0, +, −, +, −, +, −; +, −), (0, 2+, −, +, −; +, −), (0, 2+, −, 2+; 2−), (2+, −, 2+, −; 0, −), (3+, −, +, −; 0, −) and (+, −, +, −, +, −; 0, +). We will show that the configurations (0, 3+, −, +; 2−) and (0, 4+, −; 2−) are not possible. For the configuration (0, 3+, −, +; 2−) we denote by p1 , p2 , p3 the three consecutive points with positive index being p1 the closest point to p0 and by p5 the remaining point with positive index. Applying the Euler-Jacobi formula (2) to R = L0,1 L0,5 L2,3 we reach a contradiction. For the configuration (0, 4+, −; 2−) we denote by p1 , p2 , p3 , p4 the three consecutive points with positive index and by p5 , p6 , p7 the points with neg- ative index being p5 the point in the 0-level. Consider the straight line S = L6,7 . Consider the straight line L0,2 . We have three possible cases. First the points p6 , p7 are on the right-hand side of L0,2 , second the points p6 , p7 are on the left-hand side of L0,2 , and third the points p6 , p7 are one of the left-hand side, for instance p6 , and the other is on the right-hand side of L0,2 . In the first case applying (2) to L0,1 L0,2 L3,4 we have a contradiction. In the second case applying (2) to L0,1 L0,2 L4,5 we reach a contradiction, and finally in the third case applying (2) to L0,1 L0,6 L5,7 we have a contradiction. The cubic system (1) with P (x, y) = −0.63728 − 1.01236x − 1.77583y + 0.35343x2 − 1.55544xy − 0.36373y 2 − 0.29452x3 + 0.85409x2 y + y 3 , Q(x, y) = 0.05170 − 0.95142x − 0.03677y − 0.22530x2 − 0.45898xy − 0.29892y 2 + 0.15219x3 − 0.42001x2 y + xy 2 , has the singular points (1, 2), (0.58.., −0.80..), (−0.59.., 0.80..), (0.42.., −0.90..), (4, 2), (2, 2), (0.01.., −0.45..), (−0.70.., −0.70..), (−0.70.., −0.70..) in the configuration (0, −, 2+, −, +; +, −). 20 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = 2 − y − 2x2 − 2y 2 + x2 y + y 3 , Q(x, y) = −1.45224 − 1.29953x − 0.3233y + 1.58257x2 − 1.35024xy + 0.52471y 2 − 0.79129x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (1, 2), (0.03.., 2), (−0.70.., −0.70..), (0.42.., −0.90..), (−0.59.., 0.80..), (0.86.., −0.5) in the configuration (0, +, −, +, −, +, −; +, −). The cubic system (1) with P (x, y) = −0.63728 − 1.01236x − 1.77583y + 0.35343x2 − 1.55544xy − 0.36373y 2 − 0.29452x3 + 0.8541x2 y + y 3 , Q(x, y) = 0.0517 − 0.95142x − 0.03677y − 0.2253x2 − 0.45898xy − 0.29892y 2 + 0.15219x3 − 0.42001x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (−0.70.., −0.70..), (−0.70.., −0.70..), (0.42.., −0.90..), (0.01.., −0.45..), (0.58.., −0.8..), (−0.59.., 0.80..) in the configuration (0, 2+, −, +, −; +, −), The cubic system (1) with P (x, y) = −1.05986 − 1.17457x − 1.90014y + 0.73053x2 − 1.80467xy − 0.10155y 2 − 0.34171x3 + 0.83072x2 y + y 3 , Q(x, y) = 0.39154 − 0.82097x + 0.06321y − 0.52855x2 − 0.25854xy − 0.50976y 2 + 0.19014x3 − 0.40121x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (−0.70.., −0.70..), (0.42.., −0.90..), (0.41.., −0.88..), (0.58.., −0.80..), (0.58.., −0.80..), (−0.59.., 0.80..) in the configuration (0, 2+, −, 2+; 2−). The cubic system (1) with P (x, y) = −0.89569 + 1.6814x − 3.69593y + 0.46307x2 + 1.50863xy − 1.85672y 2 − 1.2361x3 + 1.89441x2 y + y 3 , Q(x, y) = −0.15127 + 1.1644x − 1.54493y − 0.13917x2 + 1.94772xy − 1.4716y 2 − 0.58739x3 + 0.39711x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 21 has the singular points (4, 2), (1, 2), (−1.18.., −0.93..), (−0.70.., −0.70..), (−0.70.., −0.70..), (−0.86.., −0.5), (0.42.., −0.90..), (0.58.., −0.80..), (0.86.., −0.5) in the configuration (2+, −, 2+, −; 0, −). The cubic system (1) with P (x, y) = 2 − y − 2x2 − 2y 2 + x2 y + y 3 , Q(x, y) = −1.31108 − 1.47454x − 0.43766y + 0.99081x2 − 1.26273xy + 0.5466y 2 − 0.49541x2 y + xy 2 , has the singular points (4, 2), (2, 2), (2, 2), (1.14.., 2), (1, 2), (−0.70.., −0.70..), (0.42.., −0.95..), (0.58.., −0.80..), (−0.59.., 0.80..) in the configuration (3+, −, +, −; 0, −). The cubic system (1) with P (x, y) = 0.22343x − 3.70599y + 1.91805x2 − 0.91189xy + 2.33258y 2 + 0.20499x3 − 1.65986x2 y + y 3 , Q(x, y) = 0.33464x − 4.37889y + 2.26518x2 − 1.91377xy + 4.02829y 2 + 0.29371x3 − 2.21375x2 y + xy 2 has the singular points (2.68.., 1.46..), (1.7, 1.3), (1.7, 1.3), (1.2, 0.64), (0.1, 1.1), (0.6, 0.2), (−0.5, 0.1), (−1.1, 0.65), (0, 0) in the configuration (+, −, +, −, +, −; 0, +). Configuration (5; 3) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, −, 2+, −; 2+, −), (0, +, −, +, −; 2+, −), (0, 2+, −, +; 2−, +), (0, 3+, −; 2−, +), (2+, −, +, −; 0, +, −) and (4+, −; 0, 2−). The cubic system (1) with P (x, y) = −169.055 + 488.263x − 94.7783y − 431.31x2 + 174.286xy − 5.32869y 2 + 116.227x3 + −63.2897x2 y + y 3 , Q(x, y) = −24.6823 + 73.049x − 14.8523y − 66.5269x2 + 29.1387xy − 1.43955y 2 + 18.7953x3 − 11.8453x2 y + xy 2 , has the singular points (2, 2), (2, 2), (1.93.., 3.44..), (0.62.., −1.07..), (1.91.., 2.85..), (0.8, 2), (0.57, −0.9), (0.58.., −0.80..), (0.86.., −0.5) in the configuration (0, −, 2+, −; 2+, −). 22 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = −0.26624 − 0.127x + 1.42367y + 0.21549x2 + 4.60997xy + 3.27449y 2 − 2.03241x3 − 2.64176x2 y + y 3 , Q(x, y) = 0.26673 + 0.74274x − 1.54738y + 0.42201x2 − 4.81204xy − 2.55609y 2 + 1.32549x3 + 1.2861x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (−0.15.., −0.41..), (−1.63.., 1.29..), (−1.63.., 1.29..), (0, 0.14), (−0.83.., 0.45..), (−0.9, 0.65) in the configuration (0, +, −, +, −; 2+, −). The cubic system (1) with P (x, y) =2 − y − 2x2 + xy − 2y 2 + x2 y + y 3 , Q(x, y) = − 1.45224 − 1.29953x − 0.3233y + 1.58257x2 − 1.35024xy + 0.52471y 2 − 0.79129x2 y + xy 2 , has the singular points (4, 2), (4, 2), (2, 2), (1, 2), (0.18.., 2), (−0.70.., −0.70..), (0.42.., −0.90..), (−0.59.., 0.80), (0.86.., −0.5) in the configuration (0, 2+, −, +; 2−, +). The cubic system (1) with P (x, y) = − 0.32744 − 0.74422x − 1.5832y + 0.38015x2 − 1.37547xy − 0.62723y 2 − 0.24965x3 + 0.68371x2 y + y 3 , Q(x, y) =0.29506 − 0.74081x + 0.11453y − 0.2043x2 − 0.31762xy − 0.50588y 2 + 0.18743x3 − 0.55384x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (−0.70.., −0.70..), (−0.70.., −0.70), (0.42.., −0.90..), (−0.59.., 0.80..), (0.40.., −0.34..), (0.86.., −0.5) in the configuration (0, 3+, −; 2−, +). The cubic system (1) with P (x, y) = − 2.65273 − 1.49182x − 3.75008y + 1.40703x2 − 4.00078xy + 1.89442y 2 − 0.6781x3 + 1.66983x2 y + y 3 , Q(x, y) = − 1.53136 − 1.32802x − 1.58746y + 0.60227x2 − 2.37969xy + 1.47477y 2 − 0.1491x3 + 0.22071x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 23 has the singular points (4, 2), (2, 2), (1, 2), (−1.077.., −0.89..), (−0.70.., −0.70..), (−0.70.., −0.70..), (−0.86.., −0.5), (−0.59.., 0.80..), (0.86.., −0.5) in the configuration (2+, −, +, −; 0, +, −). The cubic system (1) with P (x, y) = − 0.66741 − 0.85292x − 1.66839y + 0.72781x2 − 1.57638xy − 0.42671y 2 − 0.28612x3 + 0.63752x2 y + y 3 , Q(x, y) =0.4278 − 0.69837x + 0.14779y − 0.34005x2 − 0.23917xy − 0.58417y 2 + 0.20166x3 − 0.5358x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (−0.70.., −0.70..), (0.42.., −0.90..), (0.21.., −0.75..), (−0.59.., 0.80..), (0.86.., −0.5), (0.86.., −0.5) in the configuration (4+, −; 0, 2−). Configuration (4; 4) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, −, +, −; 3+, −), (0, +, −, +; +, −, +, −), (0, +, −, +; 2+, 2−), (0, 2+, −; +, −, +, −), (0, 2+, −; 2+, 2−), (0, 3+; 3−, +), (+, −, +, −; 0, +, −, +, −), (+, −, +, −; 0, 2+, −), (3+, −; 0, −, +, −), (3+, −; 0, 2−, +) and (4+; 0, 3−). The cubic system (1) with P (x, y) = 1.81818 + 0.06798x − 1.1351y + 2.57036x2 + 0.06798xy − 1.95328y 2 − 0.71989x3 − 0.75432x2 y + y 3 , Q(x, y) = 1.40791 − 0.95813x − 0.04956y + 1.71975x2 + 0.04187xy − 1.45747y 2 − 1.31898x3 + 0.09218x2 y + xy 2 , has the singular points (1.5, 2), (1.33.., 1.66..), (1.31.., 1.67..), (0. − 1), (0, −1), (0.79.., 1.3), (0.15.., −1.01), (0.53.., −1.13..), (−0.53.., 1.13..) in the configuration (0, −, +, −; 3+, −). The cubic system (1) with P (x, y) = 1.22404 + 1.65337x − 1.64431y + 1.96013x2 − 1.71128xy − 0.58091y 2 + 1.00465x3 − 1.94643x2 y + y 3 , Q(x, y) = 0.68907 + 0.70571x − 0.70214y + 2.06091x2 − 1.41422xy − 0.32726y 2 − 1.35748x3 − 0.18828x2 y + xy 2 , 24 J. LLIBRE AND C. VALLS has the singular points (1.5, 2), (1.33.., 1.66..), (−2.68.., 3.93..), (0.79.., 1.5), (0.53.., 1.13..), (−0.53.., −1.13..), (−0.53.., −1.13..), (−0.53.., 1.13..), (−0.22.., 0.78..) in the configuration (0, +, −, +; +, −, +, −). The cubic system (1) with P (x, y) = 7.55853 − 10.0716x − 1.24914y + 5.0247x2 + 10.2185xy − 7.80767y 2 − 6.16078x3 + 2.69095x2 y + y 3 , Q(x, y) = 2.06465 − 2.88002x − 0.05144y + 2.29759x2 + 1.7866xy − 2.11609y 2 − 1.76752x3 + 0.05748x2 y + xy 2 , has the singular points (1.47.., 2.02..), (1.5, 2), (1.5, 2), (1.33.., 1.66..), (1.31.., 1.67..), (0.84.., 1.3), (0.53.., 1.10..), (0, −1), (−0.53.., 1.13..) in the configuration (0, +, −, +; 2+, 2−). The cubic system (1) with P (x, y) = 0.05521 + 0.04668x − 2.22167y + 1.23859x2 − 1.12651xy − 0.9332y 2 + 0.49019x3 − 0.36319x2 y + y 3 , Q(x, y) = 0.25156 + 0.58542x − 4.02491y + 1.58971x2 − 8.43119xy − 5.19507y 2 + 3.01039x3 + 2.43304x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.6, 0.2), (−1.63.., 1.29..), (−1.63.., 1.29..), (−0.75.., 0.28..), (−1.3, 0.65), (−0.35.., 0.09..) in the configuration (0, 2+, −; +, −, +, −). The cubic system (1) with P (x, y) = −10.3655 − 8.98918x − 27.6779y + 9.36214x2 + 6.33352xy + 6.75136y 2 + 0.8147x3 + 7.51867x2 y + y 3 , Q(x, y) = 8.32828 + 3.6089x + 13.3552y − 4.42819x2 + 3.32785xy − 4.90853y 2 − 1.30944x3 − 7.55403x2 y + xy 2 , has the singular points (−2.72.., −6.63..), (−0.06.., 3.14..), (1.47.., 0.49..), (2.94.., −1.33..), (2.94.., −1.33..), (2.17.., −1.78..), (−1.09.., 0.42..), (1.09.., −0.52..), (2.35.., −1.46..) in the configuration (0, 2+, −; 2+, 2−). THE IMPROVED EULER-JACOBI FORMULA 25 The cubic system (1) with P (x, y) = 1.59672 + 0.29792x − 1.21191y + 2.50785x2 + 0.29792xy − 1.80862y 2 − 0.96331x3 − 0.81568x2 y + y 3 , Q(x, y) = 0.9278 − 0.5773x − 0.30067y + 2.22057x2 + 0.42271xy − 1.22846y 2 − 2.3668x3 + 0.26153x2 y + xy 2 , has the singular points (1.5, 2), (0, −1), (0, −1), (1.33.., 1.66..), (0.79.., 1.5), (0.53.., 1.13..), (0.53.., −1.13..), (−0.77.., 1.26..), (−0.22.., 0.78..) in the configuration (0, 3+; 3−, +). The cubic system (1) with P (x, y) = −0.66741 − 0.85292x − 1.66839y + 0.72781x2 − 1.57638xy − 0.42671y 2 − 0.28612x3 + 0.63752x2 y + y 3 , Q(x, y) = 0.4278 − 0.69837x + 0.14779y − 0.34005x2 − 0.23917xy − 0.58417y 2 + 0.20167x3 − 0.5358x2 y + xy 2 , has the singular points (4, 2), (2, 2), (1, 2), (−0.70.., −0.70..), (0.42.., −0.90..), (0.21.., −0.75..), (−0.59.., 0.80..), (0.86.., −0.5), (0.86.., −0.5) in the configuration (+, −, +, −; 0, +, −, +, −). The cubic system (1) with P (x, y) = 1.23671 − 1.23041x − 0.91629y + 2.60321x2 + 0.82175xy − 1.153y 2 + 1.03428x3 − 2.83549x2 y + y 3 , Q(x, y) = 0.70928 − 2.21598x + 0.07739y + 2.09491x2 + 0.81736xy − 0.63189y 2 − 0.00953x3 − 1.81021x2 y + xy 2 , has the singular points (1.5, 2), (1.31.., 1.67..), (0.79.., 1.3), (0.79.., 1.28..), (0, −1), (0.53.., 1.10..), (0.53.., 1.10..), (0.53.., −1.13..), (−0.53.., 1.13..) in the configuration (+, −, +, −; 0, 2+, −). The cubic system (1) with P (x, y) = 1.32167 + 1.43788x − 1.57293y + 2.16797x2 − 1.37002xy − 0.86391y 2 0.40451x3 − 1.5578x2 y + y 3 , Q(x, y) = 0.37924 + 1.38958x − 0.92865y + 1.40129x2 − 2.49725xy + 0.57086y 2 + 0.54712x3 − 1.42164x2 y + xy 2 , 26 J. LLIBRE AND C. VALLS has the singular points (1.5, 2), (1.33.., 1.66..), (0.79.., 1.5), (−0.53.., −1.13..), (0.53.., 1.13..), (−0.53.., 1.13..), (−0.22364.., 0.78..), (−0.22393.., 0.78..), (−0.22393.., −0.78..) in the configuration (3+, −; 0, −, +, −). The cubic system (1) with P (x, y) = −0.03342 + 0.13443x + 0.1563y + 0.34256x2 − 1.88158xy + 0.97155y 2 + 0.62347x3 − 1.09981x2 y + y 3 , Q(x, y) = −0.02792 + 0.01194x + 0.24347y + 0.1814x2 − 0.64693xy − 0.12559y 2 + 0.3054x3 − 0.94879x2 y + xy 2 , has the singular points (1.7, 1.3), (0.3, 0.2), (0.3, 0.2), (1.2, 0.64), (0.2, −1.2), (0.62.., 0.50..), (0.40.., 0.31..), (−0.5, 0.1), (−0.1, 0.1) in the configuration (3+, −; 0, 2−, +). The cubic system (1) with P (x, y) = 3.1825 + 0.66392x + 1.92747y + 1.30631x2 − 4.16906xy − 2.89871y 2 − 1.25514x3 + 2.09021x2 y + y 3 , Q(x, y) = −0.0862 + 0.15006x − 0.09326y + 0.0254x2 + 0.76885xy − 0.60459y 2 + 0.82558x3 − 1.92383x2 y + xy 2 , has the singular points (2, 2), (1.55.., 1.90..), (1.5, 1.8), (1.33.., 1.81..), (1.33.., 1.81..), (1, 2), (−0.86.., −0.5), (−0.70.., −0.5), (0.42.., −0.90..) in the configuration (4+; 0, 3−). Configuration (4; 3; 1) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, +, −, +; 2−, +; +), (0, +, −, +; 2+, −; −), (0, 2+, −; 2+, −; −), (0, 2+, −; 2−, +; +), (0, 3+; 3−; +), (0, 3+; 2−, +; −), (0, −, +, −; 2+, −; +), (0, −, +, −; 3+; −), (+, −, +, −; 0, +, −; +), (+, −, +, −; 0, 2+; −), (3+, −; 0, +, −; −), (3+, −; 0, 2−; +), (+, −, +, −; 2+, −; 0), (3+, −; 2−, +; 0), (4+; 3−; 0), and (4+; 0, 2−; −). No we show that the configurations (0, 2+, −; 2+, −; −) and (0, 3+; 2−, +; −) are not possible. For the configuration (0, 2+, −; 2+, −; −), we denote by p0 , p1 , p2 , p3 the points in the 0 level (being p3 the point with negative index), by p4 , p5 , p6 the points in the 1st level (being p6 the negative index) and by p7 the point in the THE IMPROVED EULER-JACOBI FORMULA 27 2nd level. Consider the straight line L6,7 . The points p1 , p2 , p4 and p5 have positive index. The line L6,7 can separate the set of points {p1 , p2 , p4 , p5 } in two ways. First one point p` is on one side of L6,7 and the other three points are on the other side of L6,7 . Second two points are on one side of L6,7 and the other two points are on the other side of L6,7 . In the first case applying (2) to L0,3 L0,` L6,7 we have a contradiction. In the second case denote by pk the closest point of the set {p1 , p2 , p4 , p5 } to L6,7 . Applying (2) to L0,3 L0,k L6,7 we reach a contradiction. For the configuration (0, 3+; 2−, +; −) denote by p0 , p1 , p2 , p3 the points in the 0 level, by p4 , p5 , p6 the points in the 1st level (being p4 with positive index) and by p7 the point in the 2nd level. Consider the straight line L0,4 . If the three singular points with negative index are in the same side of L0,4 then applying (2) to L0,1 L0,4 L2,3 we have a contradiction. If the three singular points with negative index are not all in the same side of L0,4 then applying (2) to L0,1 L0,3 L7 being L7 the straight line passing to p7 being parallel to L5,6 we reach a contradiction. The cubic system (1) with P (x, y) = 2.66278 − 2.03402x − 3.32725y + 0.13241x2 + 5.23787xy − 2.25453y 2 − 1.23728x3 − 4.87209x2 y + y 3 , Q(x, y) = 1.07346 − 1.81894x − 3.55618y + 0.48056x2 + 4.04944xy + 1.16146y 2 + 0.42202x3 + 0.58928x2 y + xy 2 , has the singular points (−2.72.., −6.63..), (2.94.., −1.33..), (2.17.., −1.78..), (−0.06.., 3.14..), (1.09.., −0.52..), (−1.09.., 0.42..), (−1.09.., 0.42..), (1.026.., −0.25..), (0.45.., 0.56..) in the configuration (0, +, −, +; 2−, +; +). The cubic system (1) with P (x, y) = 9.65409 − 1.51164x − 5.00183y + 6.8444x2 + 1.47388xy − 7.15989y 2 − 11.7997x3 + 9.61314x2 y + y 3 Q(x, y) = 1.21914 − 0.151691x + 0.432748y + 2.77906x2 − 1.75697xy − 1.74289y 2 − 2.74733x3 + 1.64549x2 y + xy 2 , has the singular points (−0.79.., 2), (0.53.., −1.13..), (1.45.., 2.14..), (1.5, 2), (1.5, 2), (1.31.., 1.67..), (0.81.., 1.3), (0.53.., 1.10..), (1.33.., 1.66..) in the configuration (0, +, −, +; 2+, −; −). 28 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = 1.58089 − 3.87837x − 8.08004y + 0.71079x2 + 5.41967xy − 0.64709y 2 − 0.58716x3 − 2.83003x2 y + y 3 Q(x, y) = −1.29881 − 0.50969x − 2.43789y + 2.54356x2 + 4.06429xy + 1.05356y 2 − 0.17975x3 + 0.78554x2 y + xy 2 , has the singular points (−2.72.., −6.63..), (2.31.., −1.98..), (2.94.., −1.33..), (2.94.., −1.33..), (2.17.., −1.78..), (−0.06.., 3.14..), (2.03.., −1.46..), (1.09.., −0.52..), (−1.09.., 0.42..) in the configuration (0, 2+, −; 2−, +; +). The cubic system (1) with P (x, y) = 1.83083 + 0.06503x − 1.12582y + 2.53496x2 + 0.06503xy − 1.95665y 2 − 0.6756x3 − 0.77083x2 y + y 3 , Q(x, y) = 1.29796 − 0.93243x − 0.13022y + 2.02728x2 + 0.06757xy − 1.42818y 2 − 1.70372x3 + 0.23551x2 y + xy 2 , has the singular points (1.5, 2), (0, −1), (0, −1), (1.38.., 1.89..), (1.31.., 1.67..), (0.79.., 1.3), (0.53.., 1.10..), (0.53.., −1.13..), (−0.53.., 1.13..) in the configuration (0, 3+; 3−; +). The cubic system (1) with P (x, y) = −19.764x − 20y − y 2 + 4.38561x2 + 3.55687xy − 0.238337x3 − 0.136263x2 y + 0.118392xy 2 , Q(x, y) = −395.28x − 400y + 109.356x2 + 113.222xy − 6.63869x3 − 5.42657x2 y + 2.58145xy 2 + y 3 , has the singular points (0, 0), (0, 0), (0, −20), (10, −9), (9.1, −6), (5, −7.87..), (9, −7.87..), (9, −7), (8.9, −7.86..) in the configuration (0, −, +, −; 2+, −; +). The cubic system (1) with P (x, y) = 1.4391 − 0.77711x − 0.99307y + 2.5933x2 + 0.5588xy − 1.43217y 2 + 0.4201x3 − 2.10839x2 y + y 3 , Q(x, y) = 0.96301 − 1.94997x + 0.11714y + 1.74667x2 + 0.61791xy − 0.84587y 2 + 0.01895x3 − 1.49701x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 29 has the singular points (1.80.., 2.37..), (1.5, 2), (1.33.., 1.66..), (1.31.., 1.67..), (0, −1), (0.79.., 1.3), (−0.53.., 1.13..), (−0.53.., 1.13..), (0.53.., −1.13..) in the configuration (0, −, +, −; 3+; −). The cubic system (1) with P (x, y) = 2.18148 + 0.82957x − 1.24948y + 2.49468x2 − 0.38158xy − 2.43096y 2 − 1.68474x3 + 0.4477x2 y + y 3 , Q(x, y) = 4.93445 + 6.99657x − 1.41267y + 1.60951x2 − 4.56412xy − 6.34712y 2 − 12.1694x3 + 12.8727x2 y + xy 2 , has the singular points (1.5, 2), (1.31.., 1.67..), (1.31.., 1.67..), (0.79.., 1.3), (0, −1), (0.53.., 1.10..), (−0.68.., 1.53..), (−0.53.., 1.13..), (0.53.., −1.13..) in the configuration (+, −, +, −; 0, +, −; +). The cubic system (1) with P (x, y) =1.2053 − 1.29832x − 0.90547y + 2.60744x2 + 0.86151xy − 1.11077y 2 + 1.12318x3 − 2.94351x2 y + y 3 , Q(x, y) =0.84068 − 2.22268x + 0.16297y + 1.75407x2 + 0.7763xy − 0.67771y 2 + 0.38682x3 − 1.93396x2 y + xy 2 , has the singular points (1.5, 2), (1.31.., 1.70..), (1.33.., 1.66..), (1.31.., 1.67..), (0, −1), (0.79.., 1.3), (0.79.., 1.3), (0.53.., −1.13..), (−0.53.., 1.13..) in the configuration (+, −, +, −; 0, 2+; −). The cubic system (1) with P (x, y) = −0.21847 − 3.23165x − 1.86475y + 8.41861x2 + 4.65148xy + 2.12508y 2 − 5.14896x3 − 3.22687x2 y + y 3 , Q(x, y) = −1.88545 − 9.51989x − 4.2883y + 29.3482x2 + 5.9392xy + 5.94989y 2 − 18.5052x3 − 1.99464x2 y + xy 2 , has the singular points (1.74.., 1.89..), (−0.2, 0.5), (1.74.., −1.89..), (0.85.., −0.10..), (0.85.., −0.10..), (−0.19.., 0.53..), (0.85.., 0.20..), (0.69.., −0.21..), (0.55.., −0.33..) in the configuration (3+, −; 0, +, −; −). 30 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = −3.98259 + 3.15619x7.96671y − 2.13359x2 + 8.36509xy− 0.53375y 2 − 0.59508x3 − −1.55334x2 y + y 3 , Q(x, y) = 1.96752 − 0.56429x + 2.74268y + 1.01136x2 − 2.82378xy − 1.28245y 2 + 0.63638x3 − 0.87949x2 y + xy 2 , has the singular points (2, 2), (1.5, 1.8), (1.33.., 1.81..), (1, 2), (1.23.., 1.84..), (−0.86.., −0.5), (−0.70.., −0.5), (−0.70.., −0.5), (0.42.., −0.90..) in the configuration (3+, −; 0, 2−; +). The cubic system (1) with P (x, y) = 1.20565 − 1.29812x − 0.90534y + 2.60678x2 + 0.8613xy − 1.11099y 2 + 1.12365x3 − 2.9434x2 y + y 3 , Q(x, y) = 0.65655 − 2.33095x + 0.09598y + 2.10096x2 + 0.88451xy − 0.56057y 2 + 0.14222x3 − 1.99345x2 y + xy 2 , has the singular points (1.5, 2), (1.31.., 1.67..), (0.79.., 1.3), (0.79.., 1.3), (0, −1), (0.52.., 1.11..), (0.53.., 1.10..), (0.53.., −1.13..), (−0.53.., 1.13..) in the configuration (+, −, +, −; 2+, −; 0). The cubic system (1) with P (x, y) = −0.70509 − 0.69397x + 4.27287y − 0.13448x2 + 4.13789xy − 3.43606y 2 − 1.67007x3 − 0.16262x2 y + y 3 , Q(x, y) = −2.11104 − 2.03548x + 12.7928y + 1 − 0.81903x2 + 2.2063xy − 9.51663y 2 − 5.57219x3 + 1.30903x2 y + xy 2 , has the singular points (1.7, 1.3), (−0.2, 0.2), (−0.2, 0.2), (−1.29.., −1.05..), (1.2, 0.64), (0.3, 0.2), (−0.5, 0.1), (−1.63.., 1.29..), (−0.8, 0.65), in the configuration (3+, −; 2−, +; 0). The cubic system (1) with P (x, y) = 1.1096 + 0.1224x − 1.02991y + 1.24418x2 − 1.13951y 2 − 0.42839x3 − 0.89532x2 y + y 3 , Q(x, y) = 0.51363 − 0.8668x − 0.03189y + 0.65715x2 − 0.54553y 2 − 1.46621x3 + 0.11163x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 31 has the singular points (1.5, 2), (0.62.., 1.20..), (0.53, 1.133), (0.534.., 1.1338..), (0.534.., 1.1338..), (−0.534.., −1.1338..), (0, −1), (0.534.., −1.1338..), (−0.534.., 1.1338..) in the configuration (4+; 3−; 0). The cubic system (1) with P (x, y) = 3.25912 + 0.55182x + 1.8978y + 1.18903x2 − 4.42068xy − 2.91515y 2 − 1.59916x3 + 2.65276x2 y + y 3 , Q(x, y) = −0.10693 + 0.18039x − 0.08524y + 0.05712x2 + 0.83693xy − 0.60014y 2 + 0.91865x3 − 2.07602x2 y + xy 2 , has the singular points (1, 2), (0.42.., −0.90..), (1.331.., 1.8113..), (1.33, 1.81), (1.336.., 1.812..), (1.336.., 1.812..), (2, 2), (−0.86.., −0.5), (−0.70.., −0.5) in the configuration (4+; 0, 2−; −). Configuration (3; 5) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, +, −; 2+, −, +, −), (0, +, −; 3+, 2−), (0, 2+; 2−, +, −, +), (0, 2+; 3−, 2+), (2+, −; 0, −, 2+, −), (2+, −; 0, +, −, +, −), (2+, −; 0, 2+, 2−), (3+; 0, 2−, +, −) and (3+; 0, 3−, +). We will show that the configurations (0, +, −; 3+, 2−), (0, 2+; 3−, 2+), (2+, −; 0, 2+, 2−) and (3+; 0, 3−, +) are not possible. For the configuration (0, +, −; 3+, 2−) denote by p0 , p1 , p2 the points in the 0 level and by p3 , p4 , p5 , p6 , p7 the points in the 1st level being p6 , p7 the points with negative index. Applying (2) to R = L0,1 L0,2 L6,7 we reach a contradiction. For the configuration (0, 2+; 3−, 2+), denote by p0 , p1 , p2 the points in the 0 level and by p3 , p4 , p5 , p6 , p7 the points in the 1st level being p6 , p7 the points with positive index. Applying (2) to R = L0,1 L0,2 L6,7 we have a contradiction. For the configuration (2+, −; 0, 2+, 2−) denote by p1 , p2 , p3 the points in the 0 level in counterclockwise sense being p3 the one with negative index, and by p0 , p4 , p5 , p6 , p7 the points in the 1st level in counterclockwise sense (being p6 , p7 with negative index). Note that there exists k1 , k2 ∈ {1, 2, 3} so the conic C = L0,6 L0,7 satisfies that C(pk1 )C(pk2 ) > 0. Note that at least pk1 or pk2 has positive index. Without loss of generality we assume it 32 J. LLIBRE AND C. VALLS is pk1 . Applying formula (2) to R = L0,6 L0,7 Lk2 k3 with k3 ∈ {1, 2, 3} so that k3 6= k1 and k3 6= k2 we reach a contradiction. For the configuration (3+; 0, 3−, +) denote by p1 , p2 , p3 the points in the 0 level and by p0 , p4 , p5 , p6 , p7 the points in the 1st level in counterclockwise sense. Consider C = L0,4 L0,7 . Note that there exists ` ∈ {1, 2, 3} so that the signs of C(p` ) and C(p5 ) are different. Applying (2) to R = CLk0 ,k1 being k0 , k1 ∈ {1, 2, 3} with k0 6= ` and k1 6= `, we get a contradiction. The cubic system (1) with P (x, y) = 0.07010 − 0.01445x − 0.73135y + 0.32004x2 + 0.70005xy + 0.60163y 2 + 0.15478x3 − 1.47483x2 y + y 3 , Q(x, y) = 0.06439 − 0.12082x − 0.54808y + 0.16133x2 + 1.65512xy − 0.45545y 2 − 0.11253x3 − 1.2832x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.2, −1.2), (−0.1, 0.1), (0.400.., 0.31..), (0.3, 0.2), (−0.5, 0.1), (0.09.., 0.12..) in the configuration (0, +, −; 2+, −, +, −). The cubic system (1) with P (x, y) = −6.8337 − 23.5185x − 3.02691y + 25.4219x2 + 25.1177xy + 4.80679y 2 − 15.2956x3 − 13.3315x2 y + y 3 , Q(x, y) = 2.77818 + 10.0337x + 0.59214y − 6.70244x2 − 11.4714xy − 2.18605y 2 + 5.10052x3 + 3.55814x2 y + xy 2 , has the singular points (1.5, 2), (1.33.., 1.66..), (−0.79.., 2), (−0.79.., 2), (0, −1), (−0.78, 1.99), (0.53.., 1.10..), (−0.67.., 1.61..), (−0.53.., 1.13..) in the configuration (0, 2+; 2−, +, −, +). The cubic system (1) with P (x, y) = 0.16718x − 0.84559y + 0.67325x2 − 0.13985xy + 0.40119y 2 − 0.07882x3 − 0.88085x2 y + y 3 , Q(x, y) = 0.2161x + 0.17696y + 0.25951x2 − 0.94753xy − 0.26355y 2 + 0.01953x3 − 0.47296x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (1.2, 0.64), (0.2, −1.2), (0, 0), (0.41.., 0.71..), (0.40.., 0.71..), (−0.5, 0.1), (−1, 0.55) in the configuration (2+, −; 0, −, 2+, −). THE IMPROVED EULER-JACOBI FORMULA 33 The cubic system (1) with P (x, y) = −1.95227 + 1.56068x − 9.84555y + 0.02429x2 + 6.40808xy − 0.55368y 2 − 0.11501x3 − 1.07423x2 y + y 3 , Q(x, y) = 1.82196 − 1.19243x − 4.12989y − 0.03222x2 + 4.35035xy − 2.42007y 2 + 0.08933x3 − 1.0964x2 y + xy 2 , has the singular points (4.82.., 2.80..), (3, 1), (−4.10.., 0), (2.60.., 0.66..), (2.51.., 0.69..), (2.53.., 0.65..), (2.55, 0.63..), (2.55.., 0.63..), (2.4, −0.8) in the configuration (2+, −; 0, +, −, +, −). The cubic system (1) with P (x, y) = 1.73639 − 0.38881x − 0.96698y + 2.16264x2 + 0.17322xy − 1.70337y 2 + 0.53463x3 − 1.68794x2 y + y 3 , Q(x, y) = 0.30263 + 3.3525x − 0.00289y + 0.01622x2 − 4.26596xy − 0.30552y 2 + 0.52779x3 + 0.1948x2 y + xy 2 , has the singular points (1.5, 2), (1.33.., 1.66..), (−0.79.., 2), (−0.78, 1.99), (0, −1), (0.53.., 1.10..), (0.29.., 1.04..), (−0.53.., 1.13..), (−0.53.., 1.13..) in the configuration (3+; 0, 2−, +, −). Configuration (3; 4; 1) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, +, −; +, −, +, −; +), (0, +, −; 2+, 2−; +), (0, +, −; 3+, −; −), (0, 2+; 3−, +; +), (0, 2+; 2+, 2−; −), (0, 2+; +, −, +, −; −), (2+, −; 0, 2−, +; +), (2+, −; 0, 2+, −; −), (2+, −; 0, −, +, −; +), (2+, −; 0, +, −, +; −), (3+; 0, 2−, +; −), (3+; 0, 3−; +), (3+; 0, −, +, −; −), (2+, −; 2+, 2−; 0), (2+, −; +, −, +, −; 0), and (3+; 3−, +; 0). We will show that the configurations (0, +, −; 2+, 2−; +), (0, 2+; 2+, 2−; −), (0, 2+; +, −, +, −; −), (2+, −; 0, −, +, −; +), (2+, −; 0, +, −, +; −), (3+; 0, −, +, −; −) and (2+, −; +, −, +, −; 0) are not possible. For the configuration (0, +, −; 2+, 2−; +), denote by p0 , p1 , p2 the points in the 0 level and by p3 , p4 , p5 , p6 the points in the first level being p3 and p4 the two consecutive points with positive index and p5 , p6 the two consecutive points with negative index. Applying formula (2) to R = L0,1 L0,2 L5,6 we reach a contradiction. For the configuration (0, 2+; 2+, 2−; −), denote by p0 , p1 , p2 the points in the 0 level and by p3 , p4 , p5 , p6 the points in the first level being p3 and p4 the two consecutive points with positive index and p5 , p6 the two consecutive 34 J. LLIBRE AND C. VALLS points with negative index. Applying formula (2) to R = L0,1 L0,2 L3,4 we reach a contradiction. For the configuration (0, 2+; +, −, +, −; −) denote by p0 , p1 , p2 the points in the 0 level, by p3 , p4 , p5 , p6 the points in the 1st level in counterclock- wise sense and by p7 the point in the 2nd level. Denote by pk1 the closest point to p0 in the set {p3 , p5 }. Denote by p` the closest point to L0,k1 in the set {p4 , p6 , p7 }. Denote by pk2 the point in {p3 , p5 } different from pk1 . There exists j1 , j2 ∈ {1, 2} so that Lj1 ,k2 leaves pj2 on one side of Lj1 ,k2 and the remaining points on the other side of Lj1 ,k2 . Applying formula (2) to L0,k1 L0,` Lj1 ,k2 we reach a contradiction. For the configuration (2+, −; 0, −, +, −; +), we denote by p1 , p2 , p3 the points in the 0 level in counterclockwise sense and by p0 , p4 , p5 , p6 the points in the first level also in counterclockwise sense. If L0,4 L0,6 leave a point with positive index p` of the 0-level (that without loss of generality we can assume that it is p1 ) then applying formula (2) to L0,4 L0,6 L2,3 we reach a contradiction. Otherwise, p` has negative index (that we assume that it is p1 ). Note that either L1,4 leaves the points p5 , p7 on the same side of L1,4 , or L1,6 leaves the points p5 , p7 on the same side of L1,4 . Without loss of generality we assume that it is L1,4 . Then there exists k1 ∈ {2, 3} so that L0,6 L0,k1 L1,4 leaves the other point in the set {p1 , p2 } on the same side of p5 , p7 . Applying formula (2) to L0,6 L0,k1 L1,4 we reach a contradiction. For the configuration (2+, −; 0, +, −, +; −), denote by p1 , p2 , p3 the points in the 0 level (being p1 , p2 with positive index and p3 with negative index) and denote by p0 , p4 , p5 , p6 the points in the 1st level in counterclockwise sense and by p7 the point in the 2nd level. There exists k0 ∈ {1, 2} so that the conic C = L0,5 L0,7 evaluated on pk0 and on p4 have the same sing, or the conic C evaluated on pk0 and on p6 have the same sign (note that C evaluated on p4 and on p6 have different sign). Applying (2) to R = CLk1 ,k2 with k1 , k2 ∈ {1, 2, 3} with k1 6= k0 and k2 6= k0 we reach a contradiction. For the configuration (3+; 0, −, +, −; −), denote by p1 , p2 , p3 the points in the 0 level, by p0 , p4 , p5 , p6 the points in the 1st level in counterclockwise sense and by p7 the point in the 2nd level. There exits k1 , k2 ∈ {1, 2, 3} so that Lk1 ,5 leaves k2 on one side of Lk1 ,5 and the rest of the points in the other side of Lk1 ,5 . Denote by k3 the remaining point in the 0 level (different from k0 and k1 ). Denote by p` the closest point to L0,k3 in the set {4, 6, 7}. Applying formula (2) to L0,k3 L0,` Lk1 ,5 we reach a contradiction. For the configuration and (2+, −; +, −, +, −; 0) denote by p1 , p2 , p3 the points in the 0 level (being p1 , p2 with positive index and p3 with negative index) by p4 , p5 , p6 , p7 the points in the 1st level in counterclockwise sense (being p4 and p6 with positive index) and by p0 the point in the 2nd level. There exists k0 ∈ {1, 2} so that the conic C = L0,5 L0,7 evaluated on pk0 and on p4 have the same sign, or the conic C evaluated on pk0 and on p6 THE IMPROVED EULER-JACOBI FORMULA 35 have the same sign (note that the conic C evaluated on p4 and on p6 have different signs). Applying (2) to R = CLk1 ,k2 with k1 , k2 ∈ {1, 2, 3} with k1 6= k0 and k2 6= k0 we reach a contradiction. The cubic system (1) with P (x, y) = −0.0629x − 0.62613y + 0.71196x2 + 0.66221xy + 0.73704y 2 − 0.2665x3 − 1.2599x2 y + y 3 , Q(x, y) = 0.50188x − 1.27453y + 0.92819x2 + 1.64485xy − 1.06428y 2 − 1.57961x3 + 0.31205x2 y + xy 2 , has the singular points (1.7, 1.3), (1.41.., 0.95..), (1.2, 0.64), (0.2, −1.2), (0.2, −1.2), (0.40.., 0.31..), (−1.63.., 1.29..), (−1.1, 0.65), (0, 0) in the configuration (0, +, −; +, −, +, −; +). The cubic system (1) with P (x, y) = 3851.43 − 7426.34x − 7061.69y + 2914.87x2 + 12386.9xy − 813.074y 2 + 589.519x3 − 4778.06x2 y + y 3 , Q(x, y) = −1388.63 + 2677.53x + 2546.87y − 1050.86x2 − 4466.74xy + 291.434y 2 − 212.616x3 + 1722.41x2 y + xy 2 , has the singular points (1.74.., 1.89..), (1.74.., 1.89..), (1.74.., −1.89..), (1.10.., 0), (0.1, 0.5), (0.85.., 0.10..), (0.85.., −0.08..), (−0.19.., 0.53..), (−0.39.., 0.54..) in the configuration (0, +, −; 3+, −; −). The cubic system (1) with P (x, y) = −0.14321 − 0.04623x − 0.28656y + 0.69539x2 + 0.70614xy + 1.11684y 2 − 0.2855x3 − 1.50591x2 y + y 3 , Q(x, y) = 0.44569 + 0.84699x − 0.88299y − 0.11115x2 − 3.36763xy − 1.88701y 2 + 0.63082x3 + 1.12932x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.40.., 0.31..), (−0.5, 0.1), (−1.63.., 1.29..), (−1.1, 0.65), (−1.1, 0.65), (−1.08.., 0.65..) in the configuration (0, 2+; 3−, +; +). 36 J. LLIBRE AND C. VALLS The cubic system (1) with P (x, y) = 0.06804x − 0.06224y + 0.27383x2 − 1.1449xy + 0.89605y 2 + 0.53725x3 − 1.23682x2 y + y 3 , Q(x, y) = 0.17249x − 0.23800y − 0.2008x2 + 0.52425xy − 0.31002y 2 − 0.91859x3 + 0.42637x2 y + xy 2 , has the singular points (1.7, 1.3), (0.2, −1.2), (−0.30.., −0.39..), (0.3, 0.2), (0.3, 0.2), (0.40.., 0.31..), (−1.63.., 1.29..), (−1.1, 0.65), (0, 0) in the configuration (2+, −; 0, 2−, +; +). The cubic system (1) with P (x, y) = 0.22436y − 0.02526x2 − 1.54369xy + 1.08593y 2 + 0.75301x3 − 1.20812x2 y + y 3 , Q(x, y) = 0.48854y − 0.959x2 − 0.4867xy + 0.17134y 2 − 0.37163x3 + 0.49914x2 y + xy 2 , has the singular points (1.7, 1.3), (1.18.., 0.99..), (0.2, −1.2), (0, 0), (0, 0), (0.40.., 0.31..), (0.3, 0.2), (−1.63.., 1.29..), (−1.1, 0.65) in the configuration (2+, −; 0, 2+, −; −). The cubic system (1) with P (x, y) = − 0.39665y + 0.53911x2 − 0.11032xy + 0.79396y 2 + 0.07436x3 − 1.25172x2 y + y 3 , Q(x, y) = − 3.10565y + 2.30737x2 + 7.80916xy − 1.5185y 2 − 4.29942x3 + 0.24677x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0, 0), (0, 0) (0.40.., 0.31..), (0.32.., 0.13..), (−1.63.., 1.29..), (−1.1, 0.65) in the configuration (3+; 0, 2−, +; −). The cubic system (1) with P (x, y) = 0.09811x − 0.282y + 0.40366x2 − 0.85096xy + 0.7576y 2 + 0.36605x3 − 1.16101x2 y + y 3 , Q(x, y) = 0.22473x + 0.2107y + 0.25304x2 − 1.27757xy − 0.29072y 2 + 0.13727x3 − 0.43116x2 y + xy 2 , THE IMPROVED EULER-JACOBI FORMULA 37 has the singular points (1.7, 1.3), (1.2, 0.64), (0, 0), (−0.5, 0.1), (−0.5, 0.1), (0.2, −1.2), (0.40.., 0.43..), (−1.1, 0.65), (1.07.., 0.58..) in the configuration (3+; 0, 3−; +). The cubic system (1) with P (x, y) = 0.1436x − 0.01373y + 0.21951x2 − 1.67597xy + 0.83957y 2 + 0.74789x3 − 1.18366x2 y + y 3 , Q(x, y) = −0.13225x + 0.19145y + 0.47895x2 − 0.57423xy − 0.15865y 2 + 0.06518x3 − 0.81562x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.3, 0.2), (0.3, 0.2), (0.56.., 0.59..), (0.40.., 0.43..), (−1.1, 0.65), (0, 0) in the configuration (2+, −; 2+, 2−; 0). The cubic system (1) with P (x, y) = −7.77285 − 26.0531x − 3.25264y + 27.9708x2 + 27.8513xy + 5.52021y 2 − 17.0304x3 − 14.6074x2 y + y 3 , Q(x, y) = 3.1688 + 11.088x + 0.68603y − 7.76257x2 − 12.6084xy − 2.48277y 2 + 5.82205x3 + 4.08885x2 y + xy 2 , has the singular points (1.5, 2), (1.33.., 1.66..), (−0.94.., 2.55..), (−0.79.., 2), (0, −1), (−0.78, 1.99), (−0.78, 1.99), (0.53.., 1.10..), (−0.53.., 1.13..) in the configuration (3+; 3−, +; 0). Configuration (3; 3; 2) Since there are four points with positive index and three points with negative index, taking into account Lemma 6 we can have only the configurations (0, +, −; 2+, −; +, −), (0, +, −; 2−, +; 2+) (0, +, −; 3+; 2−), (0, 2+; 2−, +; +, −), (0, 2+; 3−; 2+), (0, 2+; 2+, −; 2−), (2+, −; 0, +, −; +, −), (2+, −; 0, 2−; 2+), (2+, −; 0, 2+; 2−), (3+; 0, +, −; 2−), (3+; 0, 2−; +, −), (2+, −; 2−, +; 0, +), (3+; 3−; 0, +), (2+, −; 2+, −; 0, −), and (3+; 2−, +; 0, −). We will see that the configurations (0, +, −; 2−, +; 2+), (0, +, −; 3+; 2−), (0, 2+; 2+, −; 2−), (2+, −; 0, 2+; 2−), (3+; 0, +, −; 2−) and (3+; 2−, +; 0, −) are not possible. For the configuration (0, +, −; 2−, +; 2+) denote by p1 , p2 the points in the 0-level and by p3 , p4 the points in the 1st level with negative index. Applying (2) with L0,1 L0,2 L3,4 we have a contradiction. 38 J. LLIBRE AND C. VALLS For the configuration (0, 2∗; 3∗; 2−) (which in particular contains the con- figurations (0, +, −; 3+; 2−) and (0, 2+; 2+, −; 2−)) we note that there exists at least one point p` in the 0 level that has positive index. Without loss of generality we call it p1 . We denote by p0 , p1 , p2 the points in the 0 level, by p3 , p4 , p5 the points in the first level and by p6 , p7 the points in the 2nd level. There exists k1 , k2 ∈ {3, 4, 5} so that C = L0,k1 L0,k2 evaluated on p6 , p7 , p1 has the same sign. Denote by k3 the remaining index in {3, 4, 5} different from k1 , k2 . Applying (2) to L0,k1 L0,k2 L2,k3 we reach a contradiction. For the configuration (2+, −; 0, 2+; 2−) denote by p1 , p2 , p3 the points in the 0-level being p3 the negative, by p0 , p4 , p5 the points in the 1st-level and by p6 , p7 the points in the 2nd level. The two straight lines C = L0,4 L0,5 intersect the sides of the triangle with vertices p1 , p2 , p3 in four points. So one side of this triangle has two of these four points. If p3 is in the same component of R2 \ C then applying (2) with CL1,2 we get a contradiction. If p3 is not in the same component than R2 \ C there exists ` ∈ {1, 2} such that p` is in the same component of R2 \ C than p6 and p7 . Then applying (2) to CL3,` we have a contradiction. For the configuration (3+; 0, +, −; 2−) we denote by p1 , p2 , p3 the points in the 0 level, by p4 , p5 the points in the 1st level (being p4 the point with positive index) and by p6 , p7 the points in the 2nd level. We can rename p1 , p2 , p3 so that R = L0,1 L0,2 L3,4 evaluated on p5 , p6 and p7 have the same sign. Applying formula (2) to R we reach a contradiction. Finally, for the configuration (3+; 2−, +; 0, −) denote by p1 , p2 , p3 the points in the 0-level, by p4 , p5 , p6 the points in the first level (being p6 the point with positive index) and by p0 , p7 the points in the 2nd level. There exists `0 , `1 , `2 ∈ {4, 5, 7} so that C = L0,`0 L0,`1 evaluated on p6 and on p`2 have the same sign. Moreover, there exists k0 ∈ {1, 2, 3} so that C evaluated on p6 and on pk0 has different sign. Applying the Euler-Jacobi formula (2) to R = CLk1 ,k2 with k1 6= k2 , k1 , k2 ∈ {1, 2, 3} being k1 6= k0 and k2 6= k0 we reach a contradiction. The cubic system (1) with P (x, y) = −0.63917 − 0.03915x + 1.54354y + 0.0948x2 + −0.09761xy − 0.9535y 2 + 0.01618x3 − 0.20338x2 y + y 3 , Q(x, y) = −1.41632 − 0.11903x + 10.7244y + 0.19431x2 − 5.71385xy − 2.63511y 2 + 0.03392x3 + 0.57344x2 y + xy 2 , has the singular points (269.39.., −130.87..), (−4.10.., 0), (−4.10.., 0), (2.49.., −0.27..), (2.48.., 0.55..), (2.55.., 0.63..), (2.60.., 0.66..), (3, 1), (2.34.., 0.65..) in the configuration (0, +, −; 2+, −; +, −). THE IMPROVED EULER-JACOBI FORMULA 39 The cubic system (1) with P (x, y) = 1.74803 − 0.42482x − 1.08034y + 2.55396x2 + 0.46753xy − 1.82837y 2 − 0.29212x3 − 1.28883x2 y + y 3 , Q(x, y) = 0.13374 + 0.26732x + 0.08368y + 2.04637x2 − 2.03967xy − 0.05005y 2 − 0.10762x3 − 1.02755x2 y + xy 2 , has the singular points (1.63.., 2.14..), (1.5, 2), (1.33.., 1.66..), (1.31.., 1.67..), (0, −1), (0, 79.., 1.3), (−0.79.., 2), (−0.79.., 2), (0.53.., 1.10..) in the configuration (0, 2+; 2−, +; +, −). The cubic system (1) with P (x, y) = − 50.1135x − y + 11.1057x2 − 25.6451xy + 0.0683157x3 + 2.20461x2 y − 3.23766xy 2 + 0.101397y 3 , Q(x, y) =17.5763x2 + 0.082767xy + y 2 − 0.029669x3 + 3.41881x2 y + 0.0242532xy 2 + 0.318429y 3 , has the singular points (0, 0), (0, 0), (−5, −5), (7, −5), (−2, −4.5), (2, −4.5), (0, −3.14..), (−1/2, −3.5), (1/2, −3.5) in the configuration (0, 2+; 3−; 2+). The cubic system (1) with P (x, y) = 4.51755 + 4.06796x − 1.97571y + 2.2832x2 − 1.57532xy − 5.49326y 2 − 7.53714x3 + 7.1848x2 y + y 3 , Q(x, y) = 1.15404 + 1.92249x − 0.24618y + 1.94662x2 − 2.79227xy − 1.40022y 2 − 2.77672x3 + 2.09418x2 y + xy 2 , has the singular points (1.58.., 2.03..), (1.5, 2), (1.5, 2), (1.33.., 1.66..), (0, −1) (1.31.., 1.67..), (0.79.., 1.3), (−0.79.., 2), (0.53.., 1.10..) in the configuration (2+, −; 0, +, −; +, −). The cubic system (1) with P (x, y) = 1.67312x − y − 0.826899x2 + 1.90157xy − 0.987919x3 − 1.24626x2 y + 1.42219xy 2 + y 3 , Q(x, y) = −1.6012x2 − 0.222771xy + y 2 + 0.820301x3 + 0.865023x2 y − 0.850119xy 2 − y 3 , 40 J. LLIBRE AND C. VALLS has the singular points (0, 0), (0, 0), (−5, −5), (7, −5), (0, 1), (−2, −2, 1), (2, −2.1), (1.89.., −2), (−1.89.., −2) in the configuration (2+, −; 0, 2−; 2+). The cubic system (1) with P (x, y) = −4.34579 + 10.04x + 39.1074y − 22.3386x2 − 61.6263xy + 25.3103y 2 + 25.8492x3 − 9.25676x2 y + y 3 , Q(x, y) = −5.04257 + 11.6444x + 45.5646y − 25.7126x2 − 72.0976xy + 27.8699y 2 + 29.631x3 − 9.55345x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.1, 0.1), (0.1, 0.1), (0.79.., 0.23..), (0.6, 0.2), (0.7, 0.1), (−1.63.., 1.29..) in the configuration (3+; 0, 2+; +, −). The cubic system (1) with P (x, y) = 1.04785 − 3.26417x − 2.41163y + 3.37543x2 + 6.19441xy + 0.14689y 2 − 1.158x3 − 3.93093x2 y + y 3 , Q(x, y) = −0.47186 + 1.4699x + 1.8721y − 1.52x2 − 3.50447xy − 1.81296y 2 + 0.52146x3 + 1.39268x2 y + xy 2 , has the singular points (1.74.., 1.89..), (1.74.., −1.89..), (1.10.., 0), (0.90.., 0), (0.90.., 0), (0.1, 0.5), (0.850.., −0.1008..), (0.856.., −0.109..), (−0.19.., 0.53..) in the configuration (2+, −; 2−, +; 0, +). The cubic system (1) with P (x, y) = 0.06826x − 0.04133y + 0.28897x2 − 1.25411xy + 0.89334y 2 + 0.59877x3 − 1.2708x2 y + y 3 , Q(x, y) = 0.17828x + 0.30521y + 0.19261x2 − 2.31316xy − 0.38029y 2 + 0.67984x3 − 0.45642x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.3, 0.2), (0.3, 0.2), (−0.5, 0.1), (−1.1, 0.65), (−0.13.., 0.03..), (0, 0) in the configuration (3+; 3−; 0, +). THE IMPROVED EULER-JACOBI FORMULA 41 The cubic system (1) with P (x, y) =0.05945x − 0.05036y + 0.28675x2 − 1.16699xy + 0.90087y 2 + 0.58306x3 − 1.29606x2 y + y 3 , Q(x, y) =0.05071x + 0.07078y + 0.13478x2 − 0.04988xy − 0.18464y 2 + 0.27164x3 − 1.11276x2 y + xy 2 , has the singular points (1.7, 1.3), (1.2, 0.64), (0.2, −1.2), (0.3, 0.2), (0.3, 0.2), (0.62.., 0.50..), (0.40.., 0.31..), (−0.5, 0.1), (0, 0) in the configuration (2+, −; 2+, −; 0, −). Acknowledgements The first author is supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278- P (FEDER) and PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 Euro- pean Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] V. Arnold, A. Varchenko and S. Goussein-Zude, Singularités des applications différentialbes, Mir, Moscow, 1982. [2] A. N. Berlinskii, On the behavior of the integral curves of a differential equation, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (1960), 3–18. [3] R.F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Company, Glenview, IL, 1971. [4] C. Chicone and T. Jinghuang, On general properties of quadratic systems, Amer. Math. Monthly 81 (1982), 167–178. [5] A. Cima, A. Gasull and F. Manõsas, Some applications of the Euler-Jacobi for- mula to differential equations, Proc. Amer. Math. 118 (1993), 151–163. [6] A. Cima and J. Llibre, Configurations of fans and nests of limit cycles for polyno- mial vector fields in the plane, J. Differential Equations 82 (1989), 71–97. [7] A. Gasull and J. Torregrosa, Euler-Jacobi formula for double points and appli- cations to quadratic and cubic systems, Bull. Belg. Math. Soc. 6 (1999), 337–346. [8] A. G. Khovanskii, Index of a polynomial vector field, Funktsional Anal. i Prilozhen 13 (1979), 49–58. [9] J. Llibre and C. Valls, On the singularities of the planar cubic polynomial dif- ferential systems and the Euler Jacobi formula, to appear in Qualitative Theory of Dynamical Systems, (2020). [10] N.G. Lloyd, Degree Theory, Cambridge Tracks in Mathematics, 73, Cambridge University Pres, New York, 1978. 42 J. LLIBRE AND C. VALLS 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt