Cubic systems with invariant straight lines of total multiplicity eight and with three distinct infinite singularities

Transcription

1 Cubic systems with invariant straight lines of total multiplicity eight and with three distinct infinite singularities Cristina Bujac Nicolae Vulpe CRM-3331 December 2013 The second author is partially supported by the grant FP7-PEOPLE-2012-IRSES and by the grant F from SCSTD of ASM Institute of Mathematics and Computer cience, Academy of Science of Moldova, 5 Academiei str, Chişinău, MD-2028, Moldova; nvulpe@gmail.com, crisulicica@yahoo.com artes@mat.uab.cat

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3 Abstract In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess eight invariant straight lines, including the line at infinity and including their multiplicities and in addition they possess three distinct infinite singularities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of algebraic invariants and comitants The algebraic invariants and comitants allow one to verify for any given real cubic system whether or not it has invariant straight lines of total multiplicity eight, and to specify its configuration of straight lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems in this class, given in any normal form Mathematical Subject Classification. 34G20, 34A26, 14L30, 34C14, 37C15. Keywords and phrases. cubic differential system, Poincaré compactification, algebraic invariant curve, configuration of invariant straight lines, multiplicity of an invariant straight line, group action, affine invariant polynomial Résumé Dans cet article nous démontrons un théorème de classification (Main Theorem) de champs de vecteurs cubic réels dans le plan, qui possèdent huit droites invariantes incluant la droite à l infini et incluant leurs multiplicités et en plus trois points singuliers distincts à l infini. Cette classification faite par rapport à l action du groupe de transformations réelles affines et des homothéties sur l axe du temps, est donnée en termes d invariants et comitants. Les invariants et comitants algébriques nous permettent de verifier pour tout systèm donné s il a des droites invariantes de multiplicités totale huit et de spécifier sa configuration de droites invariantes munies de singularités réelles du système. Les calculs peuvent être faits à l ordinateur et les resultats peuvent donc s appliquer à toute famille de systèmes cubiques de cette classe, présentée par rapport a n importe quelle forme normale.

4 1 Introduction and the statement of the Main Theorem We consider here real polynomial differential systems dx dt = P (x, y), dy dt = Q(x, y), (1) where P, Q are polynomials in x, y with real coefficients, i.e. P, Q R[x, y]. We say that systems (1) are cubic if max(deg(p ), deg(q)) = 3. Let X = P (x, y) x + Q(x, y) y be the polynomial vector field corresponding to systems (1). A straight line f(x, y) =ux + vy + w =0,(u, v) (0, 0) satisfies X(f) =up (x, y)+vq(x, y) =(ux + vy + w)r(x, y) for some polynomial R(x, y) if and only if it is invariant under the flow of the systems. If some of the coefficients u, v, w of an invariant straight line belongs to C \ R, thenwesaythatthe straight line is complex; otherwisethe straight line is real. Note that, since systems (1) are real, if a system has a complex invariant straight line ux + vy + w =0, then it also has its conjugate complex invariant straight line ūx + vy + w =0. To a line f(x, y) = ux + vy + w = 0, (u, v) (0, 0) we associate its projective completion F (X, Y, Z) = ux + vy + wz = 0 under the embedding C 2 P 2 (C), (x, y) [x : y : 1]. The line Z =0inP 2 (C) is called the line at infinity of the affine plane C 2. It follows from the work of Darboux (see, for instance, [6]) that each system of differential equations of the form (1) over C yields a differential equation on the complex projective plane P 2 (C) which is the compactification of the differential equation Qdx Pdy =0inC 2. The line Z = 0 is an invariant manifold of this complex differential equation. Definition 1.1 ( [23]). We say that an invariant affine straight line f(x, y) =ux + vy + w =0(respectively the line at infinity Z =0) for a cubic vector field X has multiplicity m if there exists a sequence of real cubic vector fields X k converging to X, such that each X k has m (respectively m 1) distinct invariant affine straight lines f j i = u j i x + vj i y + wj i =0, (uj i,vj i ) (0, 0), (uj i,vj i,wj i ) C3, converging to f =0as k (with the topology of their coefficients), and this does not occur for m +1 (respectively m). We give here some references on polynomial differential systems possessing invariant straight lines. For quadratic systems see [7,20,21,23,25 27] and [28]; for cubic systems see [11 14,16,22,31] and [32]; for quartic systems see [30] and [34]; for some more general systems see [9, 17, 18] and [19]. According to [2] the maximum number of invariant straight lines taking into account their multiplicities for a polynomial differential system of degree m is 3m when we also consider the infinite straight line. This bound is always reached if we consider the real and the complex invariant straight lines, see [5]. So the maximum number of the invariant straight lines (including the line at infinity Z = 0) for cubic systems is 9. A classification of all cubic systems possessing the maximum number of invariant straight lines taking into account their multiplicities have been made in [12]. We also remark that a subclass of the family of cubic systems with invariant straight lines was discussed in [31] and [32]. More precisely, in these articles authors consider the cubic systems with exactly 7 invariant affine straight lines considered with their parallel multiplicity. They say that an invariant line f(x, y) =ux + vy + w = 0 of a cubic system (1) has the parallel multiplicity 1 k 3iftheidentity X(f) =f k R(x, y) holds for some polynomial R(x, y). In the paper [3] the classification of all cubic systems possessing invariant straight lines of total multiplicity 8 (including the line at infinity) and four distinct infinite singularities has been done. The goal of this paper is to classify the family of cubic systems with three distinct infinite singularities (real or complex), which possess invariant straight lines of total multiplicity 8, including the line at infinity and taking into account their multiplicities. It is well known that for a cubic system (1) there exist at most 4 different slopes for invariant affine straight lines, for more information about the slopes of invariant straight lines for polynomial vector fields, see [1]. Definition 1.2 ([28]). Consider a planar cubic system (1). We call configuration of invariant straight lines of this system, the set of (complex) invariant straight lines (which may have real coefficients) of the system, each endowed with its own multiplicity and together with all the real singular points of this system located on these invariant straight lines, each one endowed with its own multiplicity. Remark 1.1. In order to describe the various kinds of multiplicity for infinite singular points we use the concepts and notations introduced in [24]. Thus we denote by (a, b) the maximum number a (respectively b) of infinite (respectively finite) singularities which can be obtained by perturbation of the multiple point. 1

5 Assume that a cubic system (1) possesses 8 distinct invariant straight lines (including the line at infinity). We say that these lines form a configuration of type (3, 3, 1) if there exist two triplets of parallel straight lines and one additional straight line every set with different slopes. And we shall say that these lines form a configuration of type (3, 2, 1, 1) if there exist one triplet and one couple of parallel straight lines and two additional straight lines every set with different slopes. Similarly are defined configurations of types (3, 2, 2) and (2, 2, 2, 1) and these four types of the configurations exhaust all possible configurations formed by 8 invariant straight lines for a cubic system. Note that in all configurations the straight line which is omitted is the infinite one. Assume that a cubic system (1) possesses 8 invariant straight lines taking into account their multiplicities we shall say that these lines form a potential configuration of type (3, 3, 1) (respectively, (3, 2, 2); (3, 2, 1, 1); (2, 2, 2, 1)) if there exists a sequence of vector fields X k as in the definition of geometric multiplicity having 8 distinct line of type (3, 3, 1) (respectively, (3, 2, 2); (3, 2, 1, 1); (2, 2, 2, 1)). Our results are stated in the following theorem. Main Theorem. Assume that a cubic system possesses invariant straight lines of total multiplicity 8, including the line at infinity with its own multiplicity. In addition we assume that this system has three distinct infinite singularities, i.e. the conditions D 1 =0and D 3 0hold. Then I. This system has only real infinite singularities and it possesses one of the four possible configurations Config Config. 8.21, given in Figure 1, endowed with the corresponding conditions included below. If in a configuration an invariant straight line has multiplicity k > 1, then the number k appears near the corresponding straight line and this line is in bold face. Real invariant straight lines are represented by continuous lines, whereas complex invariant straight lines are represented by dashed lines. We indicate next to the real singular points of the system, located on the invariant straight lines, their corresponding multiplicities. Moreover the system could be brought via an affine transformation and time rescaling to the canonical forms, written below next to the configuration. II. This system could not have exactly 8 invariant straight lines (including the line at infinity and considered with their multiplicities) in the configuration of type (3, 3, 1) or (3, 2, 2). And this system has: A) Configuration of the type (3, 2, 1, 1) V 4 = V 5 = K 4 = K 5 = K 6 =0, K 7 0; Config L 1 0: Config L 1 =0: {ẋ = x(x 2 9hx xy y 2 ), ẏ = y 2 (9h + y), h 0; {ẋ = x(x 2 xy y 2 ), ẏ = y 3, h =0. B) Configuration of the type (2, 2, 2, 1) V 3 = K 4 = K 2 = K 8 =0, L 6 0; Config K 9 > 0: Config K 9 < 0 : {ẋ =(x 2 1)(x + y), ẏ =2x(y 2 1); {ẋ =(1+x 2 )(x + y), ẏ =2x(1 + y 2 ). The symbols (D 1, D 3,...,K 8, K 9 ) used here denote invariant polynomials defined in Section 2.1 of the paper. Figure 1: The configurations of invariant straight lines for cubic systems with 3 distinct infinite singularities 2 Preliminaries Consider real cubic systems, i.e. systems of the form: ẋ = p 0 + p 1 (x, y)+p 2 (x, y)+p 3 (x, y), ẏ = q 0 + q 1 (x, y)+q 2 (x, y)+q 3 (x, y) (2) 2

6 with real coefficients and variables x and y. The polynomials p i and q i (i =0, 1, 2, 3) are homogeneous polynomials of degree i in x and y: p 0 = a 00, p 3 (x, y) =a 30 x 3 +3a 21 x 2 y +3a 12 xy 2 + a 03 y 3, p 1 (x, y) =a 10 x + a 01 y, p 2 (x, y) =a 20 x 2 +2a 11 xy + a 02 y 2, q 0 = b 00, q 3 (x, y) =b 30 x 3 +3b 21 x 2 y +3b 12 xy 2 + b 03 y 3, q 1 (x, y) =b 10 x + b 01 y, q 2 (x, y) =b 20 x 2 +2b 11 xy + b 02 y 2. Let a = (a 00,a 10,a 01,...,a 03,b 00,b 10,b 01,...,b 03 ) be the 20-tuple of the coefficients of systems (2) and denote R[a, x, y] =R[a 00,a 10,a 01,...,a 03,b 00,b 10,b 01,...,b 03,x,y]. 2.1 The main invariant polynomials associated to configurations of invariant straight lines It is known that on the set of polynomial systems (1), in particular on the set CS of all cubic differential systems (2), acts the group Aff(2, R) of affine transformation on the plane [28]. For every subgroup G Aff(2, R) wehavean induced action of G on CS. We can identify the set CS of systems (2) with a subset of R 20 via the map CS R 20 which associates to each system (2) the 20-tuple a =(a 00,a 10,a 01,...,a 03,b 00,b 10,b 01,...,b 03 ) of its coefficients. For the definitions of an affine or GL-comitant or invariant as well as for the definition of a T -comitant and CT-comitant we refer the reader to [23]. Here we shall only construct the necessary T -comitants associated to configurations of invariant straight lines for the class of cubic systems with three distinct infinite singularities and with exactly eight invariant straight lines including the line at infinity and including multiplicities. Let us consider the polynomials C i (a, x, y) =yp i (a, x, y) xq i (a, x, y) R[a, x, y], i =0, 1, 2, 3, D i (a, x, y) = x p i(a, x, y)+ y q i(a, x, y) R[a, x, y], i =1, 2, 3. As it was shown in [29] the polynomials { C0 (a, x, y), C 1 (a, x, y), C 2 (a, x, y), C 3 (a, x, y), D 1 (a), D 2 (a, x, y) D 3 (a, x, y) } (3) of degree one in the coefficients of systems (2) are GL-comitants of these systems. Notation 2.1. Let f, g R[a, x, y] and (f,g) (k) = k ( ) k ( 1) h k f h x k h y h h=0 k g x h y k h. (f,g) (k) R[a, x, y] is called the transvectant of index k of (f,g) (cf. [8], [15]) Theorem 2.1. [33] Any GL-comitant of systems (2) can be constructed from the elements of the set (3) by using the operations: +,,, and by applying the differential operation (f,g) (k). Let us apply a translation x = x + x 0, y = y + y 0 to the polynomials p(a, x, y) andq(a, x, y). We obtain p(ã(a, x 0,y 0 ),x,y )=p(a, x + x 0,y + y 0 ), q(ã(a, x 0,y 0 ),x,y )=q(a, x + x 0,y + y 0 ). We construct the following polynomials ( (ã(a, Ω i (a, x 0,y 0 ) Res x C i x0,y 0 ),x,y ) (ã(a,,c 0 x0,y 0 ),x,y )) /(y ) i+1, Ω i (a, x 0,y 0 ) R[a, x 0,y 0 ], (i =1, 2, 3). Notation 2.2. G i (a, x, y) =Ω i (a, x 0,y 0 ) {x0=x, y 0=y} R[a, x, y] (i =1, 2, 3). Remark 2.1. We note that the constructed polynomials G 1 (a, x, y), G 2 (a, x, y) and G 3 (a, x, y) are affine comitants of systems (2) and are homogeneous polynomials in the coefficients a 00,...,b 02 and non-homogeneous in x, y and deg a G1 =3, deg a G2 =4, deg a G3 =5, deg (x,y) G1 =8, deg (x,y) G2 =10, deg (x,y) G3 =12. 3

7 Notation 2.3. Let G i (a, X, Y, Z) (i =1, 2, 3) be the homogenization of G i (a, x, y), i.e. and G 1 (a, X, Y, Z) =Z 8 G1 (a, X/Z, Y/Z), G 2 (a, X, Y, Z) =Z 10 G2 (a, X/Z, Y/Z), G 3 (a, X, Y, Z) =Z 12 G3 (a, X/Z, Y/Z), ( ) H(a, X, Y, Z) =gcd G 1 (a, X, Y, Z), G 2 (a, X, Y, Z), G 3 (a, X, Y, Z) in R[a, X, Y, Z]. The geometrical meaning of these affine comitants is given by the two following lemmas (see [12]): Lemma 2.1. The straight line L(x, y) ux + vy + w =0, u, v, w C, (u, v) (0, 0) is an invariant line for a quadratic system (2) if and only if the polynomial L(x, y) is a common factor of the polynomials G 1 (a, x, y), G 2 (a, x, y) and G 3 (a, x, y) over C, i.e. G i (a, x, y) =(ux + vy + w) W i (x, y) (i =1, 2, 3), where W i (x, y) C[x, y]. Lemma 2.2. Consider a cubic system (2) and let a R 20 be its 20-tuple of coefficients. 1) If L(x, y) ux + vy + w =0, u, v, w C, (u, v) (0, 0) is an invariant straight line of multiplicity k for this system then [L(x, y)] k gcd( G 1, G 2, G 3 ) in C[x, y], i.e. there exist W i (a, x, y) C[x, y] (i =1, 2, 3) such that G i (a, x, y) =(ux + vy + w) k W i (a, x, y), i =1, 2, 3. 2) If the line l : Z =0is of multiplicity k>1 then Z k 1 gcd(g 1, G 2, G 3 ), i.e. we have Z k 1 H(a, X, Y, Z). In order to define the needed invariant polynomials we first construct the following comitants of second degree with respect to the coefficients of the initial system: S 1 =(C 0,C 1 ) (1), S 10 =(C 1,C 3 ) (1), S 19 =(C 2,D 3 ) (1), S 2 =(C 0,C 2 ) (1), S 11 =(C 1,C 3 ) (2), S 20 =(C 2,D 3 ) (2), S 3 =(C 0,D 2 ) (1), S 12 =(C 1,D 3 ) (1), S 21 =(D 2,C 3 ) (1), S 4 =(C 0,C 3 ) (1), S 13 =(C 1,D 3 ) (2), S 22 =(D 2,D 3 ) (1), S 5 =(C 0,D 3 ) (1), S 14 =(C 2,C 2 ) (2), S 23 =(C 3,C 3 ) (2), S 6 =(C 1,C 1 ) (2), S 15 =(C 2,D 2 ) (1), S 24 =(C 3,C 3 ) (4), S 7 =(C 1,C 2 ) (1), S 16 =(C 2,C 3 ) (1), S 25 =(C 3,D 3 ) (1), S 8 =(C 1,C 2 ) (2), S 17 =(C 2,C 3 ) (2), S 26 =(C 3,D 3 ) (2), S 9 =(C 1,D 2 ) (1), S 18 =(C 2,C 3 ) (3), S 27 =(D 3,D 3 ) (2). We shall use here the following invariant polynomials constructed in [12] to characterize the family of cubic systems possessing the maximal number of invariant straight lines: D 1 (a) = [ 6S24 3 (C 3,S 23 ) (4)] 2, D 2 (a, x, y) = S 23, D 3 (a, x, y) = (S 23,S 23 ) (2) 6C 3 (C 3,S 23 ) (4), D 4 (a) = (C 3, D 2 ) (4), V 1 (a, x, y) = S 23 +2D3 2, V 2 (a, x, y) = S 26, V 3 (a, x, y) = 6S 25 3S 23 2D3 2, V 4 (a, x, y) = C 3 [(C 3,S 23 ) (4) +36(D 3,S 26 ) (2)], L 1 (a, x, y) = 9C 2 (S S 27 ) 12D 3 (S 20 +8S 22 ) 12 (S 16,D 3 ) (2) 3(S 23,C 2 ) (2) 16 (S 19,C 3 ) (2) +12(5S S 22,C 3 ) (1), L 2 (a, x, y) = 32(13S S 21,D 2 ) (1) +84(9S 11 2S 14,D 3 ) (1) +8D 2 (12S S 18 73S 20 ) 448 (S 18,C 2 ) (1) 56 (S 17,C 2 ) (2) 63 (S 23,C 1 ) (2) + 756D 3 S D 1 S (S 17,D 2 ) (1) 378 (S 26,C 1 ) (1) +9C 1 (48S 27 35S 24 ). 4

8 However these invariant polynomials are not sufficient to characterize the cubic systems with invariant straight lines of the total multiplicity 8. So we use here the following invariant polynomials, constructed in [3]: where V 5 (a, x, y) =6T 1 (9A 3 7A 4 )+2T 2 (4T 16 T 17 ) 3T 3 (3A 1 +5A 2 )+3A 2 T 4 +36T5 2 3T 44, L 5 (a) =2A 3 1 A 2 3, L 6 (a, x, y) =(T 10,T 10 )) (2), K 2 (a, x, y) =T 74, K 4 (a, x, y) =T 13 2T 11, K 5 (a, x, y) =45T 42 T 2 T 14 +2T 2 T T T 37 45T T 39, K 6 (a, x, y) =4T 1 T 8 (2663T T 15 )+6T 8 (178T T T 26 )+18T 9 (30T 2 T 8 488T 1 T T 21 )+5T 2 (25T T 137 ) 15T 1 (25T T 141 ) 165T 142, K 7 (a) =A 1 +3A 2, K 8 (a, x, y) =10A 8 T 1 3T 2 T 15 +4T 36 8T 37, K 9 (a, x, y) =3T 1 (11T 15 8T 14 ) T 23 +5T 24, A 1 = S 24 /288, A 2 = S 27 /72, A 3 = ( S 23,C 3 ) (4)/2 7 /3 5, A 4 = ( S 26,D 3 ) (2)/2 5 /3 3, A 8 = [ 9D 1 (S A 2 )+4 ( ) (2) ( ) (1) ] 9S 11 2S 14,D S18 S 20 4S 22,D 2 /2 7 /3 3, are affine invariants, whereas the polynomials T 1 = C 3, T 2 = D 3, T 3 = S 23 /18, T 4 = S 25 /6, T 5 = S 26 /72, T 6 = [ 3C 1 (D3 2 9T 3 +18T 4 ) 2C 2 (2D 2 D 3 S 17 +2S 19 6S 21 )+ +2C 3 (2D2 2 S 14 +8S 15 ) ] /2 4 /3 2, T 8 = [ 5D 2 (D T 3 18T 4 )+20D 3 S ( ) (1) ] S 16,D 3 8D3 S 17 /5/2 5 /3 3, T 9 = [ 9D 1 (9T 3 18T 4 D3 2 )+2D 2(D 2 D 3 3S 17 S 19 9S 21 )+18 ( ) (1) S 15,C 3 ] 6C 2 (2S 20 3S 22 )+18C 1 S 26 +2D 3 S 14 /2 4 /3 3, T 11 = [( D3 2 ) (2) ( ) 9T 3 +18T 4,C 2 6 D 2 (1) ( ) (1)+ 3 9T 3 +18T 4,D 2 12 S26,C 2 ] +12D 2 S (A 1 5A 2 )C 2 /2 7 /3 4, [ T 13 = 27 ( ) (2) ( ) (2) T 3,C 2 18 T4,C 2 +48D3 S ( ) (1) T 4,D 2 +36D2 S C 2 (3A 1 +17A 2 )+ ( D 2 3,C 2 ) (2) ]/2 7 /3 4, T 14 = [( 8S 19 +9S 21,D 2 ) (1) D2 (8S 20 +3S 22 )+18D 1 S C 1 A 2 ] /2 4 /3 3, T 15 =8 ( 9S 19 +2S 21,D 2 ) (1) +3 ( 9T3 18T 4 D 2 3,C 1) (2) 4 ( S17,C 2 ) (2)+ +4 ( ) (1) ( ) (2) S 14 17S 15,D 3 8 S14 + S 15,C C1 (5A 1 +11A 2 )+ +36D 1 S 26 4D 2 (S 18 +4S 22 ) ] /2 6 /3 3, T 21 = ( ) (1), T 8,C 3 T 23 = ( ) (2)/6, T 6,C 3 T 24 = ( ) (1)/6, T 6,D 3 T 26 = ( ) (1)/4, T 9,C 3 5

9 T 29 = ( T 9,C 3 ) (1)/4, T 36 = ( T 6,D 3 ) (2)/12, T 37 = ( T 9,C 3 ) (2)/12, T 38 = ( T 9,D 3 ) (1)/12, T 39 = ( T 6,C 3 ) (3)/2 4 /3 2, T 42 = ( T 14,C 3 ) (1)/2, T 44 = ( (S 23,C 3 ) (1) ) (2)/5/2 ),D 6 3 /3 3, T 74 = [ 27C 0 (9T 3 18T 4 D3) 2 2 ( + C T11 C 3 3(9T 3 18T 4 D3) 2 (2D 2 D 3 S 17 +2S 19 6S 21 ) ) T 11 C2 2 + C 2 (9T 3 18T 4 D3) 2 (8D D 1D 3 27S S 12 4S S 15 ) 54C 3 (9T 3 18T 4 D3 2 ) (2D 1 D 2 S 8 +2S 9 ) 54D 1 (9T 3 18T 4 D3 2 )S T 6 (2D 2 D 3 S 17 +2S 19 6S 21 ) ] /2 8 /3 4, T 133 = ( ) (1), T 74,C 3 T 136 = ( ) (2)/24, T 74,C 3 T 137 = ( ) (1)/6, T 74,D 3 T 140 = ( ) (2)/12, T 74,D 3 T 141 = ( ) (3)/36, T 74,C 3 T 142 = ( (T 74,C 3 ) (2) ) (1)/72,C 3 are T -comitants of cubic systems (2) (see [23] for the definition of a T -comitant). We note that these invariant polynomials are the elements of the polynomial basis of T -comitants up to degree six constructed by Iu. Calin [4]. 2.2 Preliminary results In order to determine the degree of the common factor of the polynomials G i (a, x, y) fori =1, 2, 3, we shall use the notion of the k th subresultant of two polynomials with respect to a given indeterminate (see for instance, [10], [15]). Following [12] we consider two polynomials f(z) = a 0 z n + a 1 z n a n, g(z) =b 0 z m + b 1 z m b m, in the variable z of degree n and m, respectively. We say that the k th subresultant with respect to variable z of the two polynomials f(z) andg(z) isthe(m + n 2k) (m + n 2k) determinant R (k) z (f,g) = a 0 a 1 a a m+n 2k 1 0 a 0 a a m+n 2k a a m+n 2k b b m+n 2k 3 0 b 0 b b m+n 2k 2 b 0 b 1 b b m+n 2k 1 (m k) times (n k) times in which there are m k rows of a s and n k rows of b s, and a i =0fori>n,andb j =0forj>m. For k = 0 we obtain the standard resultant of two polynomials. In other words we can say that the k th subresultant with respect to the variable z of the two polynomials f(z) andg(z) can be obtained by deleting the first and the last k rows and the first and the last k columns from its resultant written in the form (4) when k =0. The geometrical meaning of the subresultants is based on the following lemma. Lemma 2.3. (see [10], [15]). Polynomials f(z) and g(z) have precisely k roots in common (considering their multiplicities) if and only if the following conditions hold: R z (0) (f,g) =R(1) z (f,g) =R(2) z (f,g) = = R(k 1) (f,g) =0 R (k) (f,g). z z (4) 6

10 For the polynomials in more than one variables it is easy to deduce from Lemma 2.3 the following result. Lemma 2.4. Two polynomials f(x 1,x 2,..., x n ) and g(x 1,x 2,..., x n ) have a common factor of degree k with respect to the variable x j if and only if the following conditions are satisfied: R x (0) j ( f, g) =R x (1) j ( f, g) =R x (2) j ( f, g) = = R (k 1) ( f, g) =0 R x (k) j ( f, g), where R (i) x j ( f, g) =0in R[x 1,...x j 1,x j+1,...,x n ]. In paper [12] all the possible configurations of invariant straight lines are determined in the case, when the total multiplicity of these line (including the line at infinity) equals nine. For this propose in [12] there are proved some lemmas concerning the number of triplets and/or couples of parallel invariant straight lines which could have a cubic system. In [3] these results have been completed as follows: Theorem 2.2 ([3]). If a cubic system (2) possess a given number of triplets or/and couples of invariant parallel straight lines real or/and complex, then the following conditions are satisfied, respectively: (i) 2 triplets V 1 = V 2 = U 1 =0; (ii) 1 triplet and 2 couples V 3 = V 4 = U 2 =0; (iii) 1 triplet and 1 couple V 4 = V 5 = U 2 =0; (iv) 1 triplet V 4 = U 2 =0; (v) 3 couples V 3 =0; (vi) 2 couples V 5 =0. We rewrite the systems (2) in the coefficient form: x j ẋ = a + cx + dy + gx 2 +2hxy + ky 2 + px 3 +3qx 2 y +3rxy 2 + sy 3, ẏ = b + ex + fy + lx 2 +2mxy + ny 2 + tx 3 +3ux 2 y +3vxy 2 + wy 3. (5) Let L(x, y) =Ux+ Vy+ W = 0 be an invariant straight line of this family of cubic systems. Then, we have UP(x, y)+vq(x, y) =(Ux+ Vy+ W )(Ax 2 +2Bxy + Cy 2 + Dx + Ey + F ), and this identity provides the following 10 relations: Eq 1 =(p A)U + tv =0, Eq 6 =(2h E)U +(2m D)V 2BW =0, Eq 2 =(3q 2B)U +(3u A)V =0, Eq 7 = ku +(n E)V CW =0, Eq 3 =(3r C)U +(3v 2B)V =0, Eq 8 =(c F )U + ev DW =0 Eq 4 =(s C)U + Vw=0, Eq 9 = du +(f F )V EW =0, Eq 5 =(g D)U + lv AW =0, Eq 10 = au + bv FW =0. (6) It is well known that the infinite singularities (real or complex) of systems (5) are determined by the linear factors of the polynomial C 3 = yp 3 (x, y) xq 3 (x, y). So in the case of three distinct infinite singularities they are determined by one double real and either two simple real or two simple complex factors of the polynomial C 3 (x, y). Lemma 2.5 ([12]). A cubic system (5) has 3 distinct infinite singularities if and only if D 1 =0and D 3 0. The types of these singularities are determined by the following conditions: (i) all real if D 3 > 0; (ii) one real and 2 imaginary if D 3 < 0. Moreover the homogeneous cubic parts of systems (5) could be brought via a linear transformation and a time rescaling to the canonical form x =3(u +1)x 3 +(v 1)x 2 y + rxy 2, y =ux 2 y + vxy 2 + ry 3, C 3 = x 2 y(x y) in the case D 1 =0, D 3 > 0 and to the canonical form in the case D 1 =0, D 3 < 0. x =ux 3 +(v +1)x 2 y + rxy 2, y = x 3 + ux 2 y + vxy 2 + ry 3, C 3 = x 2 (x 2 + y 2 ) 7

11 3 The proof of the Main Theorem Following Lemma 2.5 we split the family of cubic systems having 3 distinct infinite singularities in two subfamilies depending on the types of these singularities and namely: systems with three real distinct and systems with one real and two complex infinite singularities. For each of the families the proof of the Main Theorem proceeds in 3 steps. First we construct the cubic homogeneous parts ( P 3, Q 3 ) of systems for which the corresponding necessary conditions provided by Theorem 2.2 in order to have a given number of triplets or/and couples of invariant parallel straight lines in the respective directions are satisfied. Secondly, taking the cubic systems ẋ = P 3, ẏ = Q 3 we perturb all quadratic and all linear as well as all constant terms (keeping the cubic parts). Then using the equations (6) we determine these terms in order to get the needed number of invariant straight lines in the needed configuration. This leads us to the next remark. Remark 3.1. If the perturbed systems have a triplet (respectively a couple) of parallel invariant straight lines in the direction Ux+ Vy=0then the respective cubic homogeneous systems with right-hand parts ( P 3, Q 3 ) necessarily have the invariant line Ux+ Vy=0of the multiplicity at least three (respectively at least two). And this line has exactly the multiplicity three (respectively two) if the direction Ux+ Vy= 0is simple, i.e. Ux+ Vy is a factor of multiplicity one in the factorization of the polynomial C 3 (x, y) over C. Thus the second step ends with the construction of the canonical systems possessing the needed configuration. The third step consists in the determination of the affine invariant conditions which are necessary and sufficient for a cubic system to belong to the family of systems (constructed at the second step) possessing the corresponding configuration of invariant straight lines. 3.1 Cubic systems with one double and two simple real roots of the polynomial C 3 (x, y) Assuming that systems (5) possess three real distinct infinite singularities (i.e. the conditions D 1 =0, D 3 > 0hold), according to Lemma 2.5 via a linear transformations they could be brought to the family of systems ẋ = p 0 + p 1 (x, y)+p 2 (x, y)+(u +1)x 3 +(v 1)x 2 y + rxy 2, ẏ = q 0 + q 1 (x, y)+q 2 (x, y)+ux 2 y + vxy 2 + ry 3 (7) with C 3 = x 2 y(x y). As we have three real distinct infinite singularities and the total multiplicity of the invariant straight lines (including the line at infinity) must be 8, then the systems above could have only one of the following four possible potential configurations of invariant straight lines: (i) (3, 3, 1); (ii) (3, 2, 2); (iii) (3, 2, 1, 1); (iv) (2, 2, 2, 1). We note that only the configurations (3, 3, 1) and (3, 2, 2) could have 8 distinct invariant straight lines, other two configurations could be only potential configurations for systems (7) (see the respective definition on the page 2), because we have only 3 distinct infinite singularities Systems with configurations (3, 3, 1) or (3, 2, 2) In the case of two triplets of parallel invariant straight lines, according to Theorem 2.2 the conditions V 1 = V 2 =0 are necessary for systems (7). For these systems we calculate: 4 2 V 1 =16 V 1j x 4 j y j, V 2 =8 V 2j x 2 j y j, j=0 j=0 where V 10 = u(3 + 2u), V 14 =2r 2, V 11 = 2u +4uv +3v, V 20 = 3v 2u, V 12 = 1 2v +4ru +3r +2v 2, V 21 =6r +4v 2, V 13 =2r( 1+2v), V 22 = 2r. Consequently, the condition V 14 =2r 2 = 0 implies r = 0 and then we obtain the following contradictory relations: V 10 = u(2u+3) = 0, 4V 20 +3V 21 = 2(4u+3) = 0. So the conditions V 1 = V 2 = 0 cannot be satisfied for systems (7). Consider now the configuration (3, 2, 2). By Theorem 2.2 the condition V 3 = V 4 = 0 is necessary for systems 4 (7). We calculate V 3 =32 V 3j x 4 j y j,wherev 30 = u(3 + u), V 31 =2u(2 v), V 32 =2+v 2ru +3r v 2, j=0 V 33 = 2r(1 + v) andv 34 = r 2. Therefore the condition V 34 = 0 implies r = 0 and then we again obtain the following contradictory relations: V 32 = (v +1)(v 2) = 0, V 4 = v(v 1)x 2 y(y x) = 0. Thus, the conditions V 3 = V 4 = 0 of Theorem 2.2 cannot be satisfied for systems (7). 8

12 3.1.2 Systems with potential configuration (3, 2, 1, 1) In this subsection we construct the cubic systems with tree real distinct infinite singularities which possess invariant straight lines with potential configuration (3, 2, 1, 1), having total multiplicity 8, as always the invariant straight line at the infinity is considered. For having the configuration (3, 2, 1, 1) a cubic system has to possess two couples of parallel invariant straight lines and, moreover, one couple must increase up to a triplet. Thus, according to Theorem 2.2, if a cubic system possesses 7 invariant straight lines in the configuration (3, 2, 1, 1), then necessarily the conditions V 4 = V 5 = U 2 =0 hold Construction of the corresponding cubic homogeneities As a first step we shall construct the cubic homogeneous parts of systems (7) for which the conditions above are fulfilled. Since we have 4 real infinite distinct singularities, according to Lemma 2.5 we consider the family of homogeneous systems ẋ =(u +1)x 3 +(v 1)x 2 y + rxy 2, ẏ = ux 2 y + vxy 2 + ry 3 (8) and we force the conditions V 4 = V 5 = U 2 = 0 to be satisfied. For systems (8) we have V 5 = 27 4 V 5j x 4 j y j,where 2 j=0 V 52 =6r 2 u, V 54 = r 2 (1 + r + v) The case r 0. Then u =0,v = (r + 1) and we have: V 4 = 18432(r 1)(1 + r)x 2 y(x y) =0, U 2 = 12288(r 1)(1 + r)(x y)y(6x 2 + r 2 xy r 2 y 2 ). So the condition V 4 = 0 implies U 2 = 0 (and in this case V 5 = 0). Thus we get the system in the case r = 1 and the system ẋ = x 3 x 2 y xy 2, ẏ = y 3 (9) ẋ = x 3 3x 2 y + xy 2, ẏ = 2xy 2 + y 3 in the case r = 1. We observe that via the transformation (x, y, t) (x, x y, t) the last system can be brought to system (9) The case r = 0. We calculate V 50 = u( 2+v)(1 + u + v) =0, V 4 = 9216(v 1)vx 2 y(y x) =0, U 2 = 12288(1 v)x 2[ u(uv 3 3u)x 2 + v(2uv 3 4u)xy + v(1 v + v 2 )y 2] and therefore we get either v =0andu(u +1)=0 orv =1andu(u + 2) = 0. We note that in both cases the condition V 5 = V 4 = 0 holds and this implies U 2 =0. 1) The subcase v =0. Then we get the system ẋ = x 2 y, ẏ = x 2 y (10) in the case u = 1 and the system ẋ = x 3 x 2 y, ẏ =0 in the case u = 0. We observe that via the transformation (x, y, t) (x, x y, t) the last system can be brought to system (10). 2) The subcase v =1. Since the additional condition u(u + 2) = 0 holds, we obtain the system in the case u = 0 and the system ẋ = x 3, ẏ = xy 2 (11) ẋ = x 3, ẏ = 2x 2 y + xy 2 in the case u = 2. We observe that via the transformation (x, y, t) (x, x y, t) the last system can be brought to system (11). Thus for the further investigation it remains three different homogeneous systems: (9), (10) and (11). We observe that for system (9) we have K 7 = 4 0, whereas for systems (10) and (11) the condition K 7 =0holds. Onthe other hand as it was shown above the condition V 4 = V 5 = 0 implies U 2 = 0. So we arrive at the following remark. Remark 3.2. If V 4 = V 5 =0then systems (8) due to a linear transformation can be brought to system (9) if K 7 0 and either to system (10) or (11) if K 7 =0. 9

13 Construction of the cubic systems possessing potential configuration (3, 2, 1, 1) In what follows we shall examine each one of the cases, when the homogeneous cubic part of systems (7) corresponds to system (9) or (10) or (11) Systems with cubic homogeneous parts (9). Considering (9) due to a translation we may assume that in the quadratic part of the cubic systems the condition g = n = 0 holds. So we consider the family of systems ẋ =a + cx + dy +2hxy + ky 2 + x 3 x 2 y xy 2, ẏ =b + ex + fy + lx 2 +2mxy y 3 (12). In what follows we shall determine necessary and sufficient conditions for a system (12) to have a potential configuration of invariant lines of the type (3, 2, 1, 1). Considering Notation 2.3 for the homogeneous system (9) we calculate H(X, Y, Z) =gcd(g 1, G 2, G 3 )=X 2 (X Y ) 2 Y 3. So the invariant line y = 0 (respectively x y = 0) of systems (9) is of multiplicity three (respectively two). Hence by Remark 3.1 the systems (12) could possess one triplet (respectively one couple) of invariant straight lines in the direction y = 0 (respectively x y = 0). As regard the direction x = 0, which is defined by the double root of C 3 (x, y), in this direction we could have one invariant straight line, having the multiplicity 2 (as it will be shown later in this direction could be a single invariant straight line). We shall examine each one of these three directions. (i) The direction y = 0. Considering the equations (6) we obtain U =0,V=1,A=0,B=0,C= 1, D=2m, E = W, F = f W 2, Eq 5 = l, Eq 8 = e 2mW, Eq 10 = b fw + W 3, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 6 = Eq 8 =0 and obviously we can have a triplet of parallel invariant straight line in the direction y =0ifandonlyifl = e = m =0. In this case the straight lines of the triplet are defined by the solutions of the equation Eq 10 = b fw + W 3 =0, which could be real or complex, distinct or coinciding. Clearly a solution W 0 of multiplicity 1 s 3 of the indicated equation define an invariant line L 0 of multiplicity s. More exactly after a perturbation of systems (9) the line L 0 splits in s distinct parallel straight lines. (ii) The direction x y =0. In this case considering the equations (6) and conditions above we have U =1,V= 1, A=1,B=0,C= 1, D = W, E =2h W, F = c + W 2, Eq 7 =2h + k, Eq 9 = c + d f 2hW +2W 2, Eq 10 = a b cw W 3, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 6 = Eq 8 =0. So we conclude that for the existence of a couple of parallel invariant straight lines for systems (12) in this direction it is necessary and sufficient k = 2h and W (Eq 9,Eq 10 )=R (1) W (Eq 9,Eq 10 )=0. We calculate R (1) W (Eq 9,Eq 10 )= 2(c d + f +2h 2 )=0 and we have d = c + f +2h 2. Then we obtain W (Eq 9,Eq 10 )=8(a b + ch + h 3 ) 2 =0 and hence we get b = a + ch + h 3. Herein we have Eq 9 =2(c + h 2 hw + W 2 ), Eq 10 = (h + W )(c + h 2 hw + W 2 ) and therefore systems (12) possess in the direction x y = 0 two parallel lines, which could be real or complex, distinct or coinciding. Thus in order to have a triplet in the direction y = 0 and a couple in the direction x y = 0, for systems (12) the following conditions are necessary and sufficient: l = e = m =0, k = 2h, d = c + f +2h 2, b = a + ch + h 3. (13) 10

14 (iii) The direction x =0. In this case considering (13) we have U =1,V=0,A=1,B= 1/2, C= 1, D= W, E =2h + W, F = c + W 2, Eq 7 = 2h + W, Eq 9 = c + f +2h 2 2hW W 2, Eq 10 = a cw W 3, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 6 = Eq 8 =0. So these equations could have only one solution, given by Eq 7 = 0, i.e. W =2h. Then we obtain Eq 9 = c + f 6h 2 =0, Eq 10 = a 2ch 8h 3 =0 and this leads to the relations f = c +6h 2 and a =2h(c +4h 2 ). So we arrive at the family of systems ẋ =(2h + x)(c +4h 2 2hx + x 2 +4hy xy y 2 ), ẏ =(3h y)(c +3h 2 +3hy + y 2 ), (14) for which we calculate (see Notation 2.3) H(a, X, Y, Z) =gcd(g 1, G 2, G 3 )=(Y 3hZ)(X +2hZ) (X 2 2XY + Y 2 + hxz hy Z + cz 2 + h 2 Z 2 )(Y 2 +3hY Z + cz 2 +3h 2 Z 2 ). Considering Lemmas 2.1 and 2.2 we conclude that systems (14) possess 6 invariant straight lines. However for having the total multiplicity 8, by Lemmas 2.1 and 2.2, the polynomial H(a, X, Y, Z) musthavedegree7inx, Y, Z. In order to find out the conditions which imply the existence of an additional invariant straight line (i.e. an additional factor in H(a, X, Y, Z)), we calculate for systems (14) the polynomials (see Notation 2.3): G 1 /H = 3X 2 2hY Z 2(c +6h 2 )Z 2 + X( Y +9hZ) =T (X, Y, Z), G 3 /H = 24(X +2hZ)(X 2 XY Y 2 2hXZ +4hY Z + cz 2 +4h 2 Z 2 ) 2 (X + Y hz). So we must impose these two polynomials to have a common factor depending on X (since we need an invariant straight line in the direction x =0). Wecalculate X (G 1/H, G 3 /H) = 2304(c +21h 2 )Z 2 (Y 2 +3hY Z + cz 2 +3h 2 Z 2 ) 3 (5Y 2 12hY Z + cz 2 +12h 2 Z 2 ) 2 and this polynomial must vanish for any values of Y and Z. Clearly this could happened if and only if c = 21h 2. In this case we get the 1-parameter family of systems ẋ = (2h + x)(17h 2 +2hx x 2 4hy + xy + y 2 ), ẏ = (3h y) 2 (6h + y), which after the translation of the origin of coordinates to the singular paint ( 2h, 3h) becomes ẋ =x( 9hx + x 2 xy y 2 ), ẏ = y 2 (9h + y). (15) Moreover we may assume h {0, 1} due to the rescaling (x, y, t) (hx, hy, t/h 2 )inthecaseh 0. Forthese systems we have H(a, X, Y, Z) = X 2 (X Y )Y 2 (X Y 9hZ)(Y +9hZ). Thus systems (15) possess 5 affine real invariant straight lines (two of them being double) of total multiplicity 7, and namely: L 1,2 = x, L 3,4 = y, L 5 = y +9h, L 6 = x y, L 7 = x y 9h. On the other hand systems (15) possess finite singularities of total multiplicity 9: M 1,2,3,4 (0, 0) M 5,6 (9h, 0), M 7 (0, 9h), M 8 ( 9h, 9h), M 9 (9h, 9h). Since h {0, 1} we get two configurations. More exactly, in the case h =1weobtainConfig and in the case h = 0 systems (15) become cubic homogeneous, possessing one triple and two double invariant straight lines. This leads to the configuration Config

15 Systems with cubic homogeneous parts (10). Considering (10) we may assume (due to a translation) that in the quadratic part of the cubic systems the condition g = m = 0 holds. So we consider the family of systems ẋ =a + cx + dy +2hxy + ky 2 x 2 y, ẏ =b + ex + fy + lx 2 + ny 2 x 2 (16) y, for which C 3 (x, y) =x 2 (x y)y. So we again have three directions, one of them being double. For these systems we could not apply Remark 3.1 because for the corresponding homogeneous cubic system (10) we have H(X, Y, Z) =gcd(g 1, G 2, G 3 )=0. However calculating this polynomial for systems (16) we obtain H(X, Y, Z) =gcd(g 1, G 2, G 3 )=Z. Therefore according to Lemma 2.2 the infinite line Z = 0 for systems (16) if of the multiplicity two. Hence it remains to find out the conditions for the existence of invariant affine straight lines of these systems of total multiplicity six. (i) The direction y = 0. Considering the equations (6) we obtain U =0,V=1,A= 1, B=0C =0,D=0,E= l, F = f + lw, Eq 5 = l + W, Eq 7 = l + n, Eq 8 = e, Eq 10 = b fw lw 2, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 6 = Eq 9 =0. Clearly we can have only one invariant straight line in this direction and for this it is necessary and sufficient e =0, l = n and R (1) W (Eq 5,Eq 10 )=b fn+ n 3 =0. The last relation gives b = n(n 2 f). (ii) The direction x y =0. In this case considering the conditions above we have U =1,V= 1, A=0,B=0,C=0,D= n, E = n +2h, F = c nw, Eq 7 =2h + k, Eq 9 = c + d f 2(h + n)w, Eq 10 = a fn+ n 3 cw + nw 2, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 6 = Eq 8 =0. (17) So the necessary condition for the existence of at least one invariant straight line is k = 2h. Moreover, we have only one straight line if h + n 0 and there are two straight lines if h = n 0andc + d f =0. We will examine these two possibilities, but first we examine the third direction. (iii) The direction x =0. In this case considering (13) and the conditions e =0,l = n and k = 2h we obtain U =1,V=0,A=1,B= 1/2, C= 1, D= W, E =2h + W, F = c + W 2, Eq 7 = 2h, Eq 9 = d 2hW W 2, Eq 10 = a cw, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 6 = Eq 8 =0. (18) We observe that for the existence of at least one invariant straight line the condition h = 0 must hold. And we could have only one invariant straight line if c 0 and we have exactly two invariant straight lines if c = a =0. Remark 3.3. We note, that if in the double direction x =0we have a single invariant straight line, then this line could be of the multiplicity at most 2 (since a perturbation could split the double infinite singularity in only two distinct singularities). If we have in this direction 2 invariant straight lines then clearly each one could be of multiplicity at most two (as at infinity we have double point). By this remark we conclude that in the direction x = 0 we could have invariant straight lines of total multiplicity at most four. As in the direction y = 0 we have exactly one straight line (so 4+1=5), then in the direction y = x there could be either one straight line or two straight lines and in the second case in the direction x =0wehave invariant straight line of total multiplicity 3. So in both cases in the direction x = 0 we must have two invariant straight lines. Thus considering (18) the condition c = a = 0 have to be satisfied and then by (17) we have 1) Assume first n 0. Thenwecalculate Eq 9 = d f 2nW =0,Eq 10 = fn+ n 3 + nw 2 =0. (19) R (1) W (Eq 9,Eq 10 )=n [ (d f) 2 4n 2 (f n 2 ) ] =0 12

16 and denoting d f = u (i.e. d = f + u) weobtainf =(4n 4 + u 2 )/(4n 2 ). Thus we arrive at the family of systems [ (2n 2 + u) 2 ẋ = 4n 2 x 2] y, ẏ =(n + y)(u 2 4n 2 x 2 +4n 3 y)/(4n 2 ), (20) for which we calculate (see Notation 2.3): H(X, Y, Z) = 1 64n 7 Z(Y + nz)(2nx 2nY + uz)(2nx 2n2 Z uz)(2nx +2n 2 Z + uz). So we observe that the degree of H as homogeneous polynomial in X, Y and Z equals 5. Since we need the degree 7, we will try to force the invariant straight lines in the direction x = 0 to be both of multiplicity two, or to increase the multiplicity of the infinite line Z =0. Wecalculate G 1 /H =n 2[ 8n 3 X 3 +8n 2 (n 2 + u)xy Z 8n 4 Y 2 Z 2nu(2n 2 + u)yz 2 u 3 Z 3 4n 2 X 2 (2nY uz) 2nX(8n 2 Y 2 u 2 Z 2 ) ], G 3 /H = 6Y 2 (2nX +2nY uz)( 2nX +2n 2 Z + uz)(2nx +2n 2 Z + uz) ( 4n 2 X 2 +4n 3 YZ+ u 2 Z 2 ). So we must impose these two polynomials to have either a common factor depending on X, or the common factor Z. Wecalculate X (G 1/H, G 3 /H) = n 33 (n 2 + u)y 11 Z 3 (Y nz)(n 2 Z ny + uz) 3 (4n 3 Y + u 2 Z) (3n 2 Y 2 + uy 2 + n 4 Z 2 + n 2 uz 2 ), Z (G 1/H, G 3 /H) = n 25 u 7 (n 2 + u)x 4 Y 11 (2n 2 X +2n 2 Y + uy )(2n 2 X +2uX +2n 2 Y + uy ) 3 (4n 6 X 2 +4n 4 ux 2 +12n 6 Y 2 +16n 4 uy 2 +7n 2 u 2 Y 2 + u 3 Y 2 ). and we observe that the first (respectively the second) polynomial vanishes for any values of Y and Z (respectively for any values of X and Y )ifandonlyu + n 2 = 0 (respectively u(u + n 2 ) = 0). However, since in the polynomial G 1 /H (respectively G 3 /H) the term which contains the maximum degree Z 3 (respectively Z 5 )is n 2 u 3 Z 3 (respectively 6u 3 (2n 2 + u) 2 Y 2 Z 5 ), we conclude that in the case u = 0 we could not obtain a new factor Z in the expression of H(X, Y, Z). So the condition u = n 2 must hold and in this case we calculate again H(X, Y, Z) = 1 64 nz(2x nz)(2x + nz)2 (Y + nz)( 2X +2Y + nz), i.e. H is of degree 6. Now we need to increase the multiplicity of one more invariant straight line. As the directions y =0andy = x are simple (i.e. are defined by the simple roots of C 3 (x, y)) it remains to try to increase multiplicity of the line 2X nz =0orZ = 0. However for systems (20) with u = n 2 we have and X (G 1/H, G 3 /H) = Y 11 (4Y + nz) 0 Z (G 1/H, G 3 /H) = n 8 XY 11 and since n 0 we obtain that both resultants could not vanish as polynomials in Y,Z (respectively in X, Y ). Thus we conclude that in the case n 0 we could not have invariant straight lines of total multiplicity 8. 2) Assume now n = 0. Then by (19) we get the relation d = f and we arrive at the family of systems ẋ =(f x 2 )y, ẏ =(f x 2 )y, which are degenerate. Thus we deduce that systems (16) could not have invariant straight lines of total multiplicity Systems with cubic homogeneous parts (11). Considering (11) we may assume that in the quadratic parts of cubic systems the condition g = m = 0 holds. So we consider the family of systems ẋ =a + cx + dy +2hxy + ky 2 + x 3, ẏ =b + ex + fy + lx 2 + ny 2 + xy 2, (21) for which C 3 (x, y) =x 2 (x y)y. 13

17 Taking into account Remark 3.1 for the homogeneous system (11) we calculate H(X, Y, Z) =gcd(g 1, G 2, G 3 )= X 5 (X Y )Y 2. So by Remark 3.1 the systems (12) could possess one triplet of invariant straight lines only in the direction x =0and one simple line in the direction y = 0. As regard a couple of invariant straight lines, there could be two possibilities: it exists ( a) in the direction y = 0 (and in this case in the direction x = 0 we have invariant straight lines of total multiplicity 4), or ( b) this couple is in the direction x = 0 (and in this case in this direction we have invariant straight lines of total multiplicity 5 and in the direction y = 0 we have a simple line). Considering configurations with 4 real distinct infinite singularities given in [3], we observe that both these possibilities could be realized using Config More exactly, the case ( a) (respectively ( b)) could be realized when the infinite singularity, being the point of intersection of the invariant affine straight lines which form a triplet, collapses with the singularity being the end of one (respectively of two) invariant straight line(s). And which one of these two situations really happens it could be determine using the equations (6). (i) The direction x =0. In this case considering systems (21) we have U =1,V=0,A=1,B=0,C=0,D= W, E =2h, F = c + W 2, Eq 7 = k, Eq 9 = d 2hW, Eq 10 = a cw W 3, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 6 = Eq 8 =0 and obviously we can have a triplet of parallel invariant straight lines (which could coincide) in the direction x = 0 if and only if k = d = h =0. (ii) The direction y =0. In this case considering the conditions above we obtain U =0,V=1,A=0,B=1/2, C=0,D= W, E = n, F = f nw, Eq 5 = l, Eq 8 = e + W 2,Eq 10 = b fw + nw 2, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 6 = Eq 7 = Eq 9 =0. So for the existence of at least one invariant straight line the condition l = 0 must holds. In this case we have a single invariant straight line in the direction y = 0 if and only if W (Eq 9,Eq 10 )=0, R (1) W (Eq 9,Eq 10 ) 0, and we have a couple of parallel invariant straight lines (which could coincide) in this direction if and only if W (Eq 9,Eq 10 )=R (1) W (Eq 9,Eq 10 )=0, where W (Eq 9,Eq 10 )=(b en) 2 + ef 2, R (1) W (Eq 9,Eq 10 )= f. (22) (iii) The direction x y =0. In this case considering the conditions determined above we have U =1,V= 1, A=1,B=1/2, C=0,D= W, E = n, F = f + nw, Eq 6 = n 2W, Eq 8 = c e f nw + W 2, Eq 10 = a b fw nw 2, Eq 1 = Eq 2 = Eq 3 = Eq 4 = Eq 5 = Eq 7 = Eq 9 =0. Clearly we could have only one straight line in this directions and this line exists if and only if W (Eq 6,Eq 8 )= W (Eq 6,Eq 10 )=0, where W (Eq 6,Eq 8 )=4c 4e 4f +3n 2, R (1) W (Eq 9,Eq 10 )=4a 4b +2fn n 3. So we obtain the relations a =(4b 2fn+ n 3 )/4, c =(4e +4f 3n 2 )/4 and considering (22) we examine two possibilities: f 0andf =0. 1) The possibility f 0. Then in the direction y = 0 could exist only one invariant straight line and as it was proved above this happens if and only if (b en) 2 + ef 2 = 0. We set a new parameter u = b en (i.e. b = u + en) and then since f 0wegete = u 2 /f 2. This leads to the family of systems 14

18 for which we calculate ẋ = 1 4f 2 (fn 2u 2fx)(2f 2 fn 2 2nu + fnx 2ux +2fx 2 ), ẏ =(u + fy)(f 2 nu ux + fny + fxy)/f 2, H(X, Y, Z) = 1 (2X 2Y nz)(fy + uz)(2fx fnz +2uZ) 16f 6 (2fX 2 + fnxz 2uXZ +2f 2 Z 2 fn 2 Z 2 2nuZ 2 ). So we observe that the degree of H as homogeneous polynomial in X, Y and Z equals 5. Since we need the degree 7 and in the direction y = 0 as well as in the direction y = x in this case we could have only one invariant straight line, we will try to force the triplet of invariant straight lines in the direction x = 0 to have a total multiplicity five. We calculate G 1 /H =f 2[ 4fn 2 YZ 2 +(3f 2 n +2fu 4n 2 u)z 3 + X 2 (4fY 4uZ)+ + X(8fnYZ +(6f 2 8nu)Z 2 ) ], G 2 /H =(fxy uxz + fnyz + f 2 Z 2 nuz 2 )(6f 2 X 3 +2f 2 X 2 Y +3f 2 nx 2 Z+ +4f 2 nxy Z +6f 3 XZ 2 3f 2 n 2 XZ 2 4u 2 XZ 2 +2f 2 n 2 YZ 2 +4f 2 uz 3 4nu 2 Z 3 ). So we must impose these two polynomials to have a common factor of degree two depending on X. By Lemma 2.4 the condition X (G 1/H, G 2 /H) =R (1) X (G 1/H, G 2 /H) =0 is necessary and sufficient. We calculate R (1) X (G 1/H, G 2 /H) =16f 8 Z 3 (fy uz) [ 6f 3 (3fn 2u)Y 3 + f 2 (6f 3 +63f 2 n 2 90fnu+32u 2 )Y 2 Z + f(81f 4 n 42f 3 u 126f 2 n 2 u + 126fnu 2 28u 3 )YZ 2 + (27f 6 81f 4 nu +36f 3 u 2 +63f 2 n 2 u 2 54fnu 3 +8u 4 )Z 3] =0 and we get that this polynomial vanishes (as polynomial in R[Y,Z]) if and only if f = 0, which contradicts to our assumption. So in the case f 0 systems (22) could not have invariant straight lines of total multiplicity 8. 2) The possibility f = 0. In this case considering (22) we get b = en and gathering all the conditions determined above we obtain g = m = d = k = h = l = f =0,b= en, a = ne + n 3 /4,c= e 3n 2 /4. These relations leads to the systems ẋ =(n + x)(4e + n 2 4nx +4x 2 )/4, ẏ =(n + x)(e + y 2 ), which are degenerate. Thus we conclude that non-degenerate systems (21) could not have invariant straight lines of total multiplicity eight Invariant conditions for the configurations Config. 8.18, 8.19 As it was shown in the previous subsection, if the conditions V 4 = V 5 = 0 a cubic system (7) could possess potential configuration of invariant straight lines (3, 2, 1, 1) only if its homogeneous cubic part via a linear transformation could be brought to the form (9). And by Remark 3.2 in this case the condition K 7 0 necessarily must hold. Moreover, for V 4 = V 5 =0andK 7 0 a cubic system (7) due to an affine transformation could be brought to the form (12). And these systems possess potential configuration of invariant straight lines of the type (3, 2, 1, 1) if and only if the following conditions are satisfied: l = m =0,k= 2h, e =0,c= 21h 2, d =8h 2, f =27h 2, a = 34h 3,b= 54h 3. (23) Following the statement A) of the Main Theorem we shall prove that these conditions are equivalent to Indeed, for systems (12) we calculate K 4 = K 5 = K 6 =0. K 4 = [ 9lx 3 +(7l 8m)x 2 y +(l +4m)xy 2 +(2h + k +2m)y 3] /9 15