This is a preprint of: “Formal Weierstrass nonintegrability criterion for some classes of polynomial differential systems in C 2 ”, Jaume Giné, Jaume Llibre, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 30(4), 2050064:1–7, 2020. DOI: [10.1142/S0218127420500649] FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION FOR SOME CLASSES OF POLYNOMIAL DIFFERENTIAL SYSTEMS IN C2 JAUME GINÉ1 AND JAUME LLIBRE2 Abstract. In this paper we present a criterion for determining the formal Weierstrass non-integrability of some polynomial differ- ential systems in the plane C2 . The criterion uses solutions of the form y = f (x) of the differential system in the plane and their asso- ciated cofactors, where f (x) is a formal power series. In particular the criterion provides necessary conditions in order that some poly- nomial differential systems in C2 be formal Weierstrass integrable. Inside this class there exists non-Liouville integrable systems. Fi- nally we extend the theory of formal Weierstrass integrability to formal Puiseux Weierstrass integrability. 1. Introduction To determine when a differential system in C2 has or has not a first integral is one of the main problems in the qualitative theory of differ- ential systems. The Liouville integrability is one of the most important theories of integrability starting with Darboux in [8, 9] and strongly de- veloped during the last decades [10, 15, 29]. This theory is based on the existence of invariant algebraic curves and their multiplicity through the exponential factors. Recently generalizations on the Liouville inte- grability theory have been done in several works, see [17, 18, 24, 25, 30]. Of course there exist differential systems which are integrable, i.e. with an explicit first integral, and that are non–Liouville integrable, an example is given below. Hence a natural question is: How to de- tect these non–Liouville integrable systems? In this work we shall see that this detection is possible for some class of polynomial differential 2010 Mathematics Subject Classification. Primary 34C05. Secondary 37C10. Key words and phrases. Weierstrass integrability, Liouville integrability, polyno- mial differential systems. 1 2 J. GINÉ AND J. LLIBRE systems through a new criterion that detects formal Weierstrass non– integrability. We also apply the criterion to some polynomial differen- tial systems. But we must start providing some preliminary definitions and results. Consider the polynomial differential system in the plane C2 (1) ẋ = P (x, y), ẏ = Q(x, y), where the functions P and Q are polynomials in the complex vari- ables x and y, i.e. P , Q ∈ C[x, y]. The degree of system (1) is m = max{deg P, deg Q}. Obviously system (1) has the associated dif- ferential equation dy Q(x, y) (2) = , dx P (x, y) and the associated vector field X = P (x, y)∂/∂x + Q(x, y)∂/∂y. We say that f (x, y) = 0 with f ∈ C[x, y], is an invariant algebraic curve of system (1), or equivalently of equation (2), if the orbital de- rivative f˙ = X f = P ∂f /∂x + Q∂f /∂y vanishes on f = 0. In particular the algebraic curve f (x, y) = 0 is invariant if by the Hilbert’s Nullstel- lensatz [3] there exists a polynomial K(x, y) ∈ C[x, y] of degree less than or equal to m − 1, called the cofactor associated to the curve f (x, y) = 0 such that ∂f ∂f (3) Xf = P +Q = Kf. ∂x ∂y A non-locally constant function H : U ⊂ C2 → C is a first integral of system (1) in the open set U if this function is constant in each solution (x(t), y(t)) of system (1) contained in U . Clearly H ∈ C 1 (U ) is a first integral of system (1) on U if and only if X H = P ∂H/∂x+Q∂H/∂y ≡ 0 on U . A function R is an integrating factor associated to a first integral H of system (1) if ∂H ∂H RP = − , and RQ = , ∂y ∂x or equivalently   ∂R ∂R ∂P ∂Q (4) P +Q =− + R = −div(X )R. ∂x ∂y ∂x ∂y FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 3 We recall that a polynomial differential system (1) has a Liouvillian first integral H if its associated integrating factor is of the form  Y D (5) R = exp Ciαi , E i where D, E and the Ci are polynomials in C[x, y] and αi ∈ C, see for more details [5, 16, 27, 29]. The functions of the form (5) are called Darboux functions. Note that the curves Ci = 0 are invariant algebraic curves of the polynomial differential system (1), and the exponential exp(D/E) is a product of some exponential factors associated to the invariant algebraic curves of system (1) or to the invariant straight line at infinity when such invariant curves have multiplicity greater than one, for more details see [4, 6, 7] or Chapter 8 of [10]. However there are differential systems without a Liouvillian first in- tegral, see for instance [11, 20, 26]. A simple example is the polynomial Liénard differential system (6) ẋ = −y + x4 , ẏ = x, that has no integrating factors of the form (5), and consequently is not Liouvillian integrable, see [11] for more details. These type of examples are included in the first generalization of the Liouvillian theory of integrability where the cofactors for non–algebraic invariant curves are defined, see [11]. The problem in this generaliza- tion is how to detect these non–algebraic invariant curves. Later on the Liouvillian theory of integrability has been extended to n–dimensional autonomous or non-autonomous differential systems in [18]. A method for detecting non–algebraic invariant curves for some polynomial dif- ferential systems was given in [12]. The next question is whether a non-algebraic invariant curve has always a polynomial cofactor. The answer to this question in general is negative, see an example given in [18]. Let C[[x]] be the set of the formal power series in the variable x with coefficients in C, and C[y] the set of the polynomials in the variable y with coefficients in C. A function of the form X̀ (7) ai (x)y i ∈ C[[x]][y] i=0 is called a formal Weierstrass polynomial in y of degree `. Here we have privileged the variable y but of course we can privileged the variable x instead of y. A formal Weierstrass polynomial whose coefficients 4 J. GINÉ AND J. LLIBRE are convergent is called a Weierstrass polynomial, for more details see [2, 17, 18]. The expression of a cofactor of an invariant curve y − g(x) = 0 where g(x) is a formal Weierstrass polynomial is given in the following result proved in [13]. Proposition 1. Let g(x) ∈ C[[x]]. An invariant curve of the form y − g(x) = 0 of a polynomial differential system (1) of degree m has a formal Weierstrass polynomial cofactor of the form (8) K(x, y) = km−1 (x)y m−1 + · · · + k1 (x)y + k0 (x). In [17] it is given a generalization of the Liouvillian integrability based in the following definition. A planar autonomous differential system is (formal) Weierstrass integrable if admits an integrating fac- tor of the form (5) where D, E and the Ci ’s are (formal) Weierstrass polynomials. Note that there are systems which are Weierstrass integrable and are not Liouvillian integrable, see for instance system (6) and example 2 below. In the papers [19, 20, 21, 23] are studied some Liénard dif- ferential systems and Abel differential equations that are Weierstrass integrable. Let C[[x, y]] be the set of all formal power series in the variables x and y with coefficients in C. The following result is given in [2]. Theorem 2. If f ∈ C[[x, y]] then it has a unique decomposition of the form Ỳ (9) f = uxr (y − gj (x)), j=1 where gj (x) are Puiseux series and r ∈ Z, r ≥ 0 and u ∈ C[[x, y]] is invertible inside the ring C[[x, y]]. We note that a Darboux integrating factor (5) is analytic function where it is defined consequently by Theorem 2 it can be written into the form (9). We say that a polynomial differential system (1) is weakly formal Weierstrass integrable if it has an integrating factor of the form Ỳ (10) R= (y − fi (x))αi , i=1 FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 5 where the functions fi (x) ∈ C[[x]]. In this work we give a criterion for detecting when a polynomial differential system (1) is not weakly formal Weierstrass integrable. We note that in general to detect the non–integrability of a differential system is a very difficult problem. 2. The criterion The criterion is based in the following result which is an extension of the results of Darboux integrability for formal invariant curves, see for instance [10]. Theorem 3. Assume that the polynomial differential system (1) is weakly formal Weierstrass integrable, that is, it has an integrating fac- tor of the form (10). If Ki is the cofactor of any formal invariant curve y − fi (x) = 0, then X̀ αi Ki = −div(X ). i=1 Proof. The existence of the integrating factor R, given in (10), forces that the curve y − fi (x) = 0, for i = 1, . . . , `, is invariant for system (1), and consequently by Proposition 4 it has a formal Weierstrass polynomial cofactor Ki such that X (y − fi (x)) = Ki (y − fi (x)). Since R is an integrating factor, from (4), we have ! Ỳ −div(X )R = X (R) = X (y − fi (x))αi i=1 X̀ Ỳ = X ((y − fj (x))αj ) (y − fi (x))αi i=1 i=1,i6=j X̀ Ỳ = αi (y − fj (x))αj −1 X (y − fj (x)) (y − fi (x))αi i=1 i=1,i6=j X̀ Ỳ = αi (y − fj (x))αj −1 Ki (y − fj (x)) (y − fi (x))αi i=1 i=1,i6=j ! X̀ Ỳ X̀ = αi Ki (y − fi (x))αi = αi Ki R. i=1 i=1 i=1 6 J. GINÉ AND J. LLIBRE This completes of the proof of the theorem.  Our criterion for detecting polynomial differential systems which are not weakly formal Weierstrass integrable works as follows. First we compute Pr the solutions of the form y = fi (x) = gi (x) + O(xr+1 ) = j j=0 aj x + O(x r+1 ) ∈ C[[x]] of the polynomial differential system (1) through the equation Eq := ẋdy/dx − ẏ = 0 up to order r in the variable x. After we compute the corresponding cofactor Li ∈ C[[x]][y] of the invariant curve y − fi (x) = 0 up to order r, using the equality X (y − gi (x)) = Li (y − gi (x)) + O(xr+1 ). From Theorem 3 if the cofactors Li ’s do not verify the equality X̀ (11) αi Li = −divr (X ) + O(xr+1 ), i=1 where divr (X ) is the divergence div(X ) up to order r in the variables x and y, then system (1) cannot be weakly formal Weierstrass integrable with fi (x) formal series. 3. Puiseux Weierstrass integrability In this section we propose a generalization of the formal Weierstrass integrability that contains all the Liouville integrability. Let C((x)) be the set of series in fractionary powers in the variable x with coefficients in C (these series are called Puiseux series), and C[y] the set of the polynomials in the variable y with coefficients in C. We call a function of the form X̀ (12) ai (x)y i ∈ C((x))[y] i=0 a Puiseux Weierstrass polynomial in y of degree `. Here we have priv- ileged the variable y but of course we can privileged the variable x instead of y. This definition is a generalization of the formal Weier- strass polynomial introduced before. The expression of a cofactor of an invariant curve y − g(x) = 0 where g(x) is a Puiseux Weierstrass polynomial is given in the following result which a generalization of Proposition 1. Proposition 4. Let g(x) ∈ C((x)). An invariant curve of the form y − g(x) = 0 of a polynomial differential system (1) of degree m has a FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 7 Puiseux Weierstrass polynomial cofactor of the form (13) K(x, y) = km−1 (x)y m−1 + · · · + k1 (x)y + k0 (x). Proof. The proof of this proposition is exactly the same than Proposi- tion 1.4 of [13].  A planar autonomous differential system is Puiseux Weierstrass in- tegrable if it admits an integrating factor of the form (5) where D, E and the Ci ’s are Puiseux Weierstrass polynomials. By definition the Puiseux Weierstrass integrable systems include the Liouville integrable systems. The criterion based in Theorem 3 is also generalizable for such inte- grable systems. In a future work we will compute the invariant solu- tions y−fi (x) of a polynomial differential system (1) when the functions fi (x) ∈ C((x)). The hard problem in this case is in what fractionary power begins the Puiseux series fi (x). 4. Examples Example 1. Now consider the polynomial differential system (14) ẋ = −y + x2 , ẏ = x + ax2 . This system has a focus at the origin because the first Poincaré-Liapunov constant is V4 = a, see for more details on the Liapunov constants the Chapter 4 of [10]. Now we study if system (14) can be weakly for- mal Weierstrass integrable, that is, with fi (x) ∈ C[[x]]. Following the criterion we propose a solution curve of the form (15) y = f (x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · up to a fixed order r in the variable x. Substituting this solution in the first ordinary differential equation Eq := ẋdy/dx − ẏ = 0 we get an infinite system of equations. In order to determine the coefficients ai up to certain order we fix the developments of f (x) and Eq. If we do that up to order 5 and we solve the finite system of equations, we find the following four solutions curves for system (14): √ 3(−5 + 1 − 8a2 ) 1 √ 1 1) y = y1 (x) = 2 + (5 + 1 − 8a2 )x2 + a(5 + (8(3 + a ) 6 9 √ 1 √ √ 1 − 8a2 )x3 + (2(−1 + 1 − 8a2 ) − a2 (9 + 1 − 8a2 ))x4 − 27 4 √ √ a(−29 − 25 1 − 8a2 + a2 (57 + 5 1 − 8a2 ))x5 + O(x6 ), 405 8 J. GINÉ AND J. LLIBRE √ 3(−5 − 1 − 8a2 ) 1 √ 1 2) y = y2 (x) = 2 + (5 − 1 − 8a2 )x2 + a(5 − (8(3 + a ) 6 9 √ 1 √ √ 1 − 8a2 )x3 + (2(−1 − 1 − 8a2 ) − a2 (9 + 1 − 8a2 ))x4 − 27 4 √ √ a(29 − 25 1 − 8a2 + a2 (−57 + 5 1 − 8a2 ))x5 + O(x6 ), 405 3) y = y3 (x) = −ix−i(i+a)x2 /3+i(2−ia+a2 )x3 /18−i(−8i+6a− 9ia2 +5a3 )x4 /270+i(−12−76ia+3a2 −66ia3 +25a4 )x5 )/3240+ O(x6 ), 4) y = y4 (x) = ix + i(−i + a)x2 /3 − i(2 + ia + a2 )x3 /18 + i(8i + 6a + 9ia2 +5a3 )x4 /270−i(−12+76ia+3a2 +66ia3 +25a4 )x5 )/3240+ O(x6 ). Next we compute their Weierstrass polynomial cofactors up to order 5 and we find that 1 1) K1 = x(135+135ax−24x2 −108a2 x2 +116ax3 −228a3 x3 )+ √ 81  1 − 8a2 x (−27 − 27ax − 24x2 + 12a2 x2 − 100ax3 + 20a3 x3 ) + O(x6 ), 1 2) K2 = x(135+135ax−24x2 −108a2 x2 +116ax3 −228a3 x3 )− √ 81  2 2 2 2 3 3 3 1 − 8a x(−27 − 27ax − 24x + 12a x − 100ax + 20a x ) + O(x6 ), 1  3) K3 = 2(1080x + 270ax2 − 192x3 − 216a2 x3 + 190ax4 + 3240 165a3 x4 ) + i(3240 + 2160ax − 1080x 2 2 2  − 540a x + 288ax + 3 240a3 x3 + 60x4 − 15a2 x4 − 125a4 x4 ) + O(x6 ), 1  4) K4 = 2(1080x + 270ax2 − 192x3 − 216a2 x3 + 190ax4 + 3240 165a3 x4 ) + i(−3240 − 2160ax + 1080x 2 2 2  + 540a x − 288ax − 3 240a3 x3 − 60x4 + 15a2 x4 + 125a4 x4 ) + O(x6 ). Now we try to see if there is a linear combination of them equal to minus the divergence, that is (16) c1 K1 + c2 K2 + c3 K3 + c4 K4 = −div5 X + O(x6 ), and this equation does not have any solution. Hence by the criterion system (14) has not an integrating factor of the form (10). Conse- quently system (14) is not weakly formal Weierstrass integrable. FORMAL WEIERSTRASS NON-INTEGRABILITY CRITERION 9 When a = 0 equation (16) has the solution 5 c1 = c2 = − (1 + c4 ) c3 = 1 + c4 . 2 In fact for a = 0 the differential system (14) is a time-reversible system because is invariant by the the symmetry (x, y, t) → (−x, y, −t). Hence it has a center at the origin. Moreover it has a Darboux integrating factor of the form R = (1 + 2y − 2x2 )−1 . So for a = 0 system (14) is Liouvillian integrable. Example 2. Consider the differential system (6). This system is time-reversible because it is invariant under the symmetry (x, y, t) → (−x, y, −t). Hence it has a center at the origin of coordinates. Moreover it has not a Liouvillian first integral, see [11]. Now we are going to apply the criterion to detect if system (6) has an integrating factor of the form (10) with fi (x) ∈ C[[x]]. P We propose a solution curve of the form y = 4j=0 aj xj + O(x5 ) and substituting this solution into the differential equation Eq := ẋdy/dx− ẏ = 0 we get an infinite system of equations. If we stop this system taking into account the equations up to order 4, and solve this finite system of equations we obtain the following five solutions curves: 1) y = y1 (x) = −ix + x4 /5 + O(x5 ), 2) y = y2 (x) = ix + x4 /5 + O(x5 ), 3) y = y3 (x) = 1/2(−3)1/3 + (−1)2/3 x2 /31/3 + x4 /3 + O(x5 ), 4) y = y4 (x) = −31/3 /2 + x2 /31/3 + x4 /3 + O(x5 ), 5) y = y5 (x) = −1/2(−1)2/3 31/3 − (−1)1/3 x2 /31/3 + x4 /3 + O(x5 ). Now we are going to compute the Weierstrass polynomial cofactors of these solution curves up to order 4. Since system (6) is of degree 4 the Weierstrass polynomial cofactors are of the form K = k0 (x) + k1 (x)y + k2 (x)y 2 + k3 (x)y 3 . Applying equation (3) to the invariant solutions y −yk (x) = O(x5 ) for k = 1, . . . , 5, we obtain for each of these solutions curves their Weierstrass polynomial cofactor up to order 4, i.e. 1) K1 = (−5i + 4x3 )/5 + O(x5 ), 2) K2 = (5i + 4x3 )/5 + O(x5 ), 3) K3 = (37/6 ix − 32/3 x + 4x3 )/3 + O(x5 ), 4) K4 = 2(32/3 x + 2x3 )/3 + O(x5 ). 5) K5 = (−37/6 ix − 32/3 x + 4x3 )/3 + O(x5 ), 10 J. GINÉ AND J. LLIBRE Now we study if system (6) satisfies equation (11). Solving the equa- tion c1 K1 + c2 K2 + c3 K3 + c4 K4 + c5 K5 = −div4 X + O(x5 ), we get the solution c3 = c4 and √ √ 15i − 5 3 + 6ic4 − 2 3c4 + 10ic5 c1 = − √ , 5(−i + 3) √ √ √ 10 3 + 4 3c4 + 5ic5 + 5 3c5 c2 = √ . 5i − 5 3 Consequently system (6) could have an integrating factor of the form (10) with fi (x) ∈ C[[x]]. We claim that this system is Weierstrass integrable because it has the integrating factor (17) R(x, y) = f2−2 (x, y). Indeed, if Ai(z) and Bi(z) are the two independent Airy functions, see a definition in [1], then we know that system system (6) is integrable because it has the first integral H(x, y) = f1 f2−1 , where f1 (x, y) = −(−2)1/3 x2 Ai((−2)2/3 y) + Ai0 ((−2)2/3 y), f2 (x, y) = (−2)1/3 x2 Bi((−2)2/3 y) − Bi0 ((−2)2/3 y). The non–algebraic curves fi (x, y) = 0 are invariant curves by the flow of system (6), satisfying the corresponding equation (3) for the vector field X associated to system (6) with cofactors K1 = K2 = 2x3 . Con- sequently it has the integrating factor given in (17), which is which is not of the form (5). In short the polynomial differential system (6) is formal Weierstrass integrable but not Liouvillian integrable. Acknowledgements The first author is partially supported by a MINECO/ FEDER grant number MTM2017-84383-P and an AGAUR (Generalitat de Catalu- nya) grant number 2017SGR-1276. 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Zhang, Integrability of dynamical systems: algebra and analysis, in Devel- opments in Mathematics 47, Sringer, 2017. 1 Departament de Matemàtica, Inspires Research Centre, Universi- tat de Lleida, Avda. Jaume II, 69; 25001 Lleida, Catalonia, Spain Email address: gine@matematica.udl.cat 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat