This is a preprint of: “Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems”, Jaume Giné, Jaume Llibre, Mathematics, vol. 10, 209:1–15, 2022. DOI: [10.3390/math10020209] INVARIANT ALGEBRAIC CURVES OF GENERALIZED LIÉNARD POLYNOMIAL DIFFERENTIAL SYSTEMS JAUME GINÉ1 AND JAUME LLIBRE2 Abstract. In this note we focus on the invariant algebraic curves of gener- alized Liénard polynomial differential systems x′ = y, y ′ = −fm (x)y − gn (x) where the degrees of the polynomials f and g are m and n respectively, and we correct some results previously stated. 1. Introduction and statement of the main results In this work we study the generalized Liénard polynomial differential systems of the form (1) x′ = y, y ′ = −fm (x)y − gn (x), where the degrees of the polynomials f and g are given by the subscripts m and n respectively. Consider F (x, y) = 0 an invariant algebraic curve of the differential system (1) where F (x, y) is a polynomial, then there exists a polynomial K(x, y) such that ∂F ∂F (2) y+ (−fm (x)y − gn (x)) = KF ∂x ∂y The knowledge of the algebraic curves of system (1) allows to study the modern Darboux and Liouvillian theories of integrability, see [6] and references therein. In fact the existence of invariant algebraic curves is a measure of integrability in such theories. Another problem is finding a bound on the degree of irreducible invariant algebraic curves of system (1). This problem goes back to Poincaré for any differential system and is known as the Poincaré problem. In 1996 Hayashi [8] stated the following result. Theorem 1. The generalized Liénard polynomial differential system (1) with fm ̸≡ 0 and m+1 ≥ n has an invariant algebraic curve if and only if there is an invariant curve y − P (x) = 0 satisfying gn (x) = −(fm (x) + P ′ (x))P (x), whereR x P (x) or P (x) + F (x) is a polynomial of degree at most one, such that F (x) = 0 f (s)ds. Given P and Q polynomials, an algebraic curve of the form (y+P (x))2 −Q(x) = 0 is called hyperelliptic curve, see for instance [9, 13, 14, 17]. In such works and others the hyperelliptic curves are used to determine algebraic limit cycles of the generalized Liénard systems (1). 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. Liénard differential systems, invariant algebraic curve, first integrals. 1 2 J. GINÉ AND J. LLIBRE Theorem 1 is also announced in [16] where the author seems not aware that the theorem is false. Theorem 1 is not correct as the following proposition shows. More precisely it shows the existence of hyperelliptic invariant algebraic curves for the generalized Liénard systems (1). Proposition 2. Under the assumptions of Theorem 1 the generalized Liénard poly- nomial differential system (1) has the following hyperelliptic invariant algebraic curves: (a) F (x, y) = −(b + ax)λ + (y − b − ax)2 = 0 for f0 (x) = −3a/2 and g1 (x) = a(b + ax − λ)/2 with a ̸= 0. (b) F (x, y) = −Ax2 + (y − ax)2 for f0 (x) = −2a and g1 (x) = (a2 − A)x with aA ̸= 0. (c) F (x, y) = −bc/(2a) − cx − acx2 /(2b) + (b + ax − y)2 = 0 for f0 (x) = −2a and g1 (x) = (2ab − c)(b + ax)/(2b) with ab ̸= 0. Proposition 2 is proved in section 2. In fact the correct statement of Theorem 1 is the following. Theorem 3. The generalized Liénard polynomial differential system (1) with fm ̸≡ 0 and m + 1 ≥ n has the invariant algebraic curve y − P (x) = 0 if gn (x) = −(fm (x) + P ′ (x))PZ(x), being P (x) or P (x) + F (x) a polynomial of degree at most x one, where F (x) = f (s)ds. 0 Theorem 3 is proved in section 2. Note that the mistake in the statement of Theorem 1 is the claim that the unique invariant algebraic curves are of the form y − P (x) = 0. Demina in [3] also detected that Theorem 1 was not correct. She found coun- terexamples to Theorem 1 with invariant algebraic curves of degree 2 and 3 in the variable y. Singer in [15] found the characterization of the systems that are Liouvillian in- tegrable. Christopher [2] rewrite this result stating that if a polynomial differential system in R2 has an inverse integrating factor of the form  Y p D (3) V = exp F αi , E i=1 i where D, E and Fi are polynomials in C[x, y] and αi ∈ C, then this differential system is Liouvillian integrable. For a definition of (inverse) integrating factor see for instance section 8.3 of [6]. We say that exp(g/h), with g and h ∈ C[x, y], is an exponential factor of the polynomial differential system (1) if there exists a polynomial L(x, y) of degree at most d where d = max{m, n − 1} such that ∂ exp(g/h) ∂ exp(g/h) y+ (−fm (x)y − gn (x)) = K exp(g/h). ∂x ∂y More information on exponential factors can be found in section 8.5 of [6]. GENERALIZED LIÉNARD POLYNOMIAL SYSTEMS 3 The existence of an inverse integrating factor (3) for a polyomial differential Ppin R is equivalent to the existence of λi and µi ∈ C not all zero such 2 system Pq that i=1 λi Ki + j=1 µj Lj = div(P, Q), where Ki and Li are the cofactors of some invariant algebraic curves and exponential factors of the given polynomial differential system, respectively. See for more details statement (iv) of Theorem 8.7 of [6]. We remark that the two kind of invariant algebraic curves mentioned in Theorem 3 can appear simultaneously in some generalized Liénard polynomial differential systems (1) as the following example shows which already appeared in [7]. The generalized polynomial Liénard differential system (4) x′ = y, y ′ = −ex3 − e2 /3x − (3x2 + 4e/3)y, has the invariant algebraic curves f1 = y + ex/3 = 0 and f2 = y + x3 + ex/3 = 0. Moreover system (4) is Liouvillian integrable because it has the inverse integrating 1/3 factor V = f1 f2 . Let U be an open subset of R2 . A C 1 function H : U → R is a first integral of system (1) if it is constant on the orbits of the system contained in U , or equivalently if dH ∂H ∂H (5) = y+ (−fm (x)y − gn (x)) = 0 on U . dt ∂x ∂y Consider W as an open subset of R2 × R. A C 1 function I : W → R is a Darboux invariant of system (1) if it is constant on the orbits of the system contained in W , or equivalently if dI ∂I ∂I (6) + y+ (−fm (x)y − gn (x)) = 0 on W . dt ∂x ∂y Pp Pq Moreover, given λi and µi ∈ C not all zero such that i=1 λi Ki + j=1 µj Lj = −s for some s ∈ C \ {0}, then the (multivalued) funtion Yp Yq   µj αi gj (7) I= Fi exp exp(st) i=1 j=1 hj is a Darboux invariant of the differential system, see for more details statement (vi) of Theorem 8.7 of [6]. Under the assumptions of Theorem 3 there are generalized Liénard polynomial differential systems (1) which are Liouvillian integrable as it is shown in the next result. Proposition 4. Under the assumptions of Theorem 3 if the generalized Liénard polynomial differential system (1) has the invariant algebraic curve y − P (x) = 0. Then the following statements hold. (a) If P (x) = −F (x) + ax + b, then system (1) has the Darboux invariant (y − P (x))eat . (b) If P (x) = b, then system (1) is Liouvillian integrable with the first integral H = ey+F (x) (y − b)b if b ̸= 0, and the first integral H = y + F (x) if b = 0. 4 J. GINÉ AND J. LLIBRE Proposition 4 is proved in section 2. We note that Proposition 4 shows that Theorem 2 of [7] and Theorem 4 of [11] are not correct because their proofs are based in the wrong Theorem 1. Proposition 5. Consider the generalized Liénard polynomial differential system (1). Let P (x) be a polynomial, then y − P (x) = 0 is an invariant algebraic curve of system (1) if and only if gn (x) = −(fm (x) + P ′ (x))P (x). Proposition 5 is proved in section 2. In fact the statement of Proposition 5 already appears in [17] without proof. Note that in Proposition 5 there are no restrictions on the degrees of the poly- nomials fm , gn and P (x). The Liouvillian integrability of the generalized Liénard polynomial differential system has been studied by several authors. The main result of [10] is that under the restriction 2 ≤ n ≤ m, then system (1) has a Liouvillian first integral if and only if gn (x) = afm (x), where a ∈ C, see also [1] for a shorter proof. Later on it was studied the Liouvillian integrability of the differential systems (1) having hyperelliptic curves of the form (y + Q(x)P (x))2 − Q(x)2 = 0, see [12]. In summary, the Liouvillian integrability in the case n > m is still open. In fact the characterization of the invariant algebraic curves of system (1) for this case is not complete. Recently it has been solved the case m = 1 and n = 2, see [5]. The case n = m + 1 is the still objective of several recent works. Thus, for instance in [3, 4] some particular cases for m = 2 and n = 3 have been solved. 2. Proofs Proof of Proposition 2. Assume that system (1) has the hyperelliptic invariant curve F = (y + P (x))2 − Q(x) = 0. Then from (2) denoting by K = K(x, y) the cofactor of F = 0 we get 2gn (x)P (x) + K(−P (x)2 + Q(x)) + y(−2gn (x) + 2P (x)(K + fm (x) + P ′ (x)) − Q ′ (x)) − y 2 (K + 2fm (x) + 2P ′ (x)) = 0. From this equality we see that K = K(x) = −2(fm (x) + P ′ (x)), P (x)Q ′ (x) 1 P 2 (x)Q ′ (x) fm (x) = −P ′ (x) − , and gn (x) = − Q ′ (x) + , 2Q(x) 2 2Q(x) where f m(x) and gn (x) must be polynomials. If we assume that deg P = p and deg Q = q, we get that deg fm = p − 1 and deg gn = max{q − 1, p2 − 1}. Since m + 1 ≥ n we obtain p ≥ max{q − 1, p2 − 1} which implies p = 1. Consequently 1 ≥ q − 1, which implies q = 1, 2. If q = 1 then P (x) = ax + b with a ̸= 0 and Q(x) must be proportional to P (x), that is, Q(x) = λP (x). So fm = −3a/2 and gn = a(ax + b − λ)/2, and F = (b + ax − y)2 − (b + ax)λ. So statement (a) follows. GENERALIZED LIÉNARD POLYNOMIAL SYSTEMS 5 If q = 2 then we have P (x) = ax + b and Q(x) = Ax2 + Bx + C with aA ̸= 0, and since fm must be a polynomial we obtain that fm = −2a, and in order that gm be a polynomial we obtain either b = B = C = 0, or A = aB/(2b) and C = (bB)/(2a) with ab ̸= 0. If b = B = C = 0 then gn = (a2 − A)x and F = −Ax2 + (y − ax)2 . Therefore statement (b) is proved. If A = aB/(2b) and C = (bB)/(2a) then g(x) = (2ab − B)(b + ax)/(2b) and F (x, y) = −bB/(2a) − Bx − aBx2 /(2b) + (b + ax − y)2 . Renaming B by c we get statement (c). □ Proof of Proposition 5. First we suppose that gn (x) = −(fm (x) + P ′ (x))P (x), and we shall prove that y − P (x) = 0 is an invariant algebraic curve. From equation (7) we have −P ′ (x)y − yfm (x) − gn (x) = K(y − P (x)). Substituting gn (x) we obtain −P ′ (x)(y − P (x)) − fm (x)(y − P (x)) = K(y − P (x)). Dividing by y − P (x) the previous equality we get K = −P ′ (x) − fm (x), which is a cofactor of degree p − 1 + m of system (1). Note that the degree of the polynomial Liénard differential system is the degree of, i.e. the maximum of m + p and 2p − 1. Now we assume that y − P (x) is an invariant algebraic curve of system (1) with cofactor K, then from (7) we have −P ′ (x)y − yfm (x) − gn (x) = K(y − P (x)). From this equality we obtain that K = K(x), then we have (K(x) + fm (x) + P ′ (x))y = K(x)P (x) − gn (x). Therefore K(x) = −(fm (x) + P ′ (x)) and gn (x) = K(x)P (x). Hence gn (x) = −(fm (x) + P ′ (x))P (x), and the proposition is proved. □ Proof of Theorem 3. By Proposition 5 we only need to prove that P (x) or P (x) + F (x) are polynomial of degree at most one. Since m + 1 ≥ n and n is the maximum of m + p and 2p − 1 where p is the degree of the polynomial P (x), we have that m + 1 ≥ m + p, consequently p ≤ 1, and the theorem is proved. □ Proof of Proposition 4. We have system (1) with gn (x) = −(fm (x) + P ′ (x))P (x) and with the invariant algebraic curve y − P (x) = 0 being P (x) = −F (x) + ax + b. Then using equation (2) we obtain that the cofactor of the invariant algebraic curve y − P (x) = 0 is K = −a. Consequently system (1) has the Darboux invariant (7), which in our case becomes I = (y − P (x))eat . Hence statement (a) is proved. Assume now that P (x) = b. Therefore g(x) = −bf (x) and the differential system becomes ẋ = y and ẋ = −(y + b)f (x), which has the Darboux first integral H = ey+F (x) (y − b)b if b ̸= 0, and the Darboux first integral H = y + F (x) if b = 0, as it is easy to check using (6). □ 6 J. GINÉ AND J. LLIBRE Acknowledgements The author is partially supported by the Agencia Estatal de Investigación grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] G. Chèze, T. 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