This is a preprint of: “Polynomial vector fields on the Clifford torus”, Jaume Llibre, Adrian Murza, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 31(4), 2150057:1–5, 2021. DOI: [10.1142/S0218127421500577] POLYNOMIAL VECTOR FIELDS ON THE CLIFFORD TORUS JAUME LLIBRE AND ADRIAN C. MURZA Abstract. First we characterize all the polynomial vector fields in R4 which have the Clifford torus as an invariant surface. After we study the number of invariant meridians and parallels that such polynomial vector fields can have in function of the degree of these vector fields. 1. Introduction and statement of the main results The Clifford torus { } 1 2 1 T = (x1 , x2 , x3 , x4 ) ∈ R : x1 + x2 = , x3 + x4 = 4 2 2 2 2 2 in geometric topology is the simplest and most symmetric Euclidean space embedding of the cartesian product of two circles. It lives in R4 , as opposed to R3 . In MathSciNet at July 22 of 2017 it appears with the keyword “Clif- ford torus” 430 references. The more recent reference is [7]. In the reference [6] are studied the meridians of the surfaces of revolution and some information about the meridians of the Clifford torus can be found there. In the references [2, 12] are studied the parallels of the surfaces of revolutions and again contains some information on the parallels of the Clifford torus. In this paper first we shall study the polynomial vector fields of arbi- trary degree in R4 having the Clifford torus invariant by their flow, and after we shall compute the maximal number of parallels and meridi- ans that a polynomial vector field of a given degree can exhibit on the Clifford torus. The maximum number of invariant hyperplanes that a polynomial vector field in Rn can have in function of its degree was given in [8]. The 2010 Mathematics Subject Classification. Primary: 34C07, 34C05, 34C40. Key words and phrases. invariant parallels, invariant meridians, polynomial vec- tor field, Clifford torus. 1 http://www.gsd.uab.cat 2 J. LLIBRE AND A.C. MURZA analogous result for the invariant straight lines of polynomial vector fields in R2 was provided before in [1]. The study of the maximum number of meridians and parallels for a torus in R3 were studied in [9], and for an algebraic torus in [10]. In surfaces of revolution in R3 the meridians and parallels invariant by polynomial vector fields have been studied in [5]. As usual we denote by R[x1 , x2 , x3 , x4 ] the ring of the polynomials in the variables x1 , x2 , x3 and x4 with real coefficients. By definition a polynomial differential system in R4 is a system of the form dxi (1) = Pi (x1 , x2 , x3 , x4 ), for i = 1, 2, 3, 4, dt where Pi (x1 , x2 , x3 , x4 ) ∈ R[x1 , x2 , x3 , x4 ]. If mi is the degree of the polynomial Pi , then m = max{m1 , m2 , m3 , m4 } is the degree of the polynomial differential system (1). We denote by 4 ∑ ∂ (2) X = Pi (x1 , x2 , x3 , x4 ) , i=1 ∂xi the polynomial vector field associated to the differential polynomial system (1) of degree m. An invariant algebraic hypersurface for the polynomial differential system (1) or for the polynomial vector field (2) is an algebraic surface f = f (x1 , x2 , x3 , x4 ) = 0 with f ∈ R[x1 , x2 , x3 , x4 ], such that for some polynomial K ∈ R[x1 , x2 , x3 , x4 ] we have (3) X f = Kf. Therefore if a solution curve of system (1) has a point on the algebraic hypersurface f = 0, then the whole solution curve is contained in f = 0. The polynomial K is called the cofactor of the invariant algebraic hypersurface f = 0. We remark that if the polynomial system has degree m, then any cofactor has at most degree m − 1. If f = 0 and g = 0 are two invariant algebraic hypersurfaces by the polynomial vector field X , then S = {f = 0} ∩ {g = 0} is an invariant algebraic surface by the vector field X , i.e. if an orbit of X has a point on the algebraic surface S, then the whole orbit is contained in S. In the next theorem we characterize all the polynomial differential systems having the Clifford torus T as an invariant algebraic surface. POLYNOMIAL VECTOR FIELDS ON THE CLIFFORD TORUS 3 Theorem 1. If the polynomial differential system (1) has the Clifford torus T invariant, then P1 = A(x21 + x22 − 1/2) − 2Cx2 , P2 = B(x21 + x22 − 1/2) + 2Cx1 , (4) P3 = D(x23 + x24 − 1/2) − 2F x4 , P4 = E(x23 + x24 − 1/2) + 2F x3 , where A, B, C, D, E and F are arbitrary polynomials in the variables x1 , x2 , x3 and x4 . Theorem 1 is proved in section 2. For all (a, b) ∈ R2 such that a2 + b2 = 1/2 a meridian of the Clifford torus T is M(a,b) = {(x1 , x2 , a, b) ∈ R4 : x21 + x22 = 1/2}, and a parallel is P(a,b) = {(a, b, x3 , x4 ) ∈ R4 : x23 + x24 = 1/2}. So a meridian is defined by the three equations x21 + x22 = 1/2, x3 = a, x4 = b with a2 + b2 = 1/2, and similarly by a parallel. In the next theorem we provide the maximum number of invariant meridians or parallels that a polynomial vector field X on T can have in function of its degree. See section 3 for the definition of the multiplicity of a meridian and of a parallel. Theorem 2. Let X be a polynomial vector field on the Clifford torus T of degree m = (m1 , m2 , m3 , m4 ) with m1 > m2 > m3 > m4 > 0. (a) The number of invariant meridians of X is at most m4 −2 taking into account their multiplicities if m4 > 3, and 4 if m4 = 3. These upper bounds are reached. (b) The number of invariant parallels of X is at most m2 − 2 taking into account their multiplicities if m2 > 3, and 4 if m2 = 3. These upper bounds are reached. Theorem 2 is proved in section 3. 2. Proof of Theorem 1 For proving Theorem 1 we shall need some definitions and results for the polynomial differential systems in R2 . 4 J. LLIBRE AND A.C. MURZA Consider the following polynomial differential system in R2 ẋ = P (x, y), (5) ẏ = Q(x, y), and let ∂ ∂ Y = P (x, y) + Q(x, y) , ∂x ∂y its associated polynomial vector field. Let f (x, y) a polynomial. Then the algebraic curve f (x, y) = 0 is invariant by system (5) if there exists a polynomial k = k(x, y) such that Yf = kf . The next result is proved in Lemma 6 of [4]. Lemma 3. Assume that the polynomial system (5) has an invariant algebraic curve f (x, y) = 0 without singularities (i.e. there are no points at which f and its first derivatives are all vanish). If (fx , fy ) = 1 (i.e. the polynomials fx and fy has no common factors), then P = Af − Cfy , Q = Bf + Dfx , where A, B and C are arbitrary polynomials in the variables x and y. Proof of Theorem 1. We consider polynomial vector fields X given in (2) of degree m in R4 having the Clifford torus T as an invariant alge- braic surface, i.e. both hypersurfaces x21 + x22 = 1/2 and x23 + x24 = 1/2 are invariant by X . Let f = x21 + x22 − 1/2 = 0 and g = x23 + x24 − 1/2 = 0. By Lemma 3 and from the definition of the invariant algebraic hypersurface f = 0 given in (3) it follows that P1 = A(x21 + x22 − 1/2) − 2Cx2 , (6) P2 = B(x21 + x22 − 1/2) + 2Cx1 , where A, B and C arbitrary polynomials in the variables x1 , x2 , x3 and x4 . In a similar way, from Lemma 3 and the definition of the invariant algebraic hypersurface g = 0 we get P3 = D(x23 + x24 − 1/2) − 2F x4 , (7) P4 = E(x23 + x24 − 1/2) + 2F x3 , where D, E and F arbitrary polynomials in the variables x1 , x2 , x3 and x4 . In short from (6) and (7) the theorem follows.  POLYNOMIAL VECTOR FIELDS ON THE CLIFFORD TORUS 5 3. Proof of Theorem 2 One of the best tools for working with invariant algebraic hypersur- faces is the extactic polynomial of X associated to a finite vector space of polynomials generated by W . To our knowledge the extactic polyno- mial was introduced by Lagutinskii, see [11]. We recall its definition for a polynomial vector field in R4 . Let W be a finitely generated vector subspace of the vector space C[x1 , x2 , x3 , xd ] generated by the basis {v1 , . . . , vl }. The extactic polynomial of X associated to W is   v1 v2 ... vl  X (v1 ) X (v2 ) . . . X (vl )  EW (X ) = E{v1 ,...,vl } (X ) = det   . .. .. .. ,  . ... . X l−1 (v1 ) X l−1 (v2 ) . . . X l−1 (vl ) where X j (vi ) = X j−1 (X (vi )). The extactic polynomial does not de- pendent of the chosen basis of W . The extactic polynomial EW (X ) has two good properties. First, it allows to detect invarian algebraic hypersurfaces f = 0 with f ∈ W by the polynomial vector field X , see the following proposition proved in [3]. Second, it allows to compute the multiplicity of the invariant algebraic hypersurfaces. Even if the next proposition is stated for complex polynomial vector fields, it is very useful for our later considerations. This is so, because we deal with real polynomial vector fields, which are particular cases of complex ones. Proposition 4. Let X be a polynomial vector field in C4 and let W be a finitely generated vector subspace of C[x1 , x2 , x3 , x4 ] with dim(W ) > 1. Then every algebraic invariant hypersurface f = 0 for the vector field X , with f ∈ W , is a factor of the polynomial EW (X ). From Proposition 4 it follows that f = 0 is an invariant hyperplane of the polynomial vector field X if the polynomial f is a factor of the polynomial EW (X ), where W is generated by {1, x1 , x2 , x3 , x4 }. From [3] the invariant hypersurface f = 0, with f ∈ W , has multi- plicity k if k is the greatest positive integer such that f k divides the polynomial EW (X ). In [3] it is proved that if we have that f = 0 is an invariante hypersurface of multiplicity k, then in a neighborhood of X in the topology of the coefficients there are polynomial vector fields Yε , being ε a small parameter, having k invariant algebraic hypersurfaces such that all of them tend to the hypersurface f = 0 when ε → 0. 6 J. LLIBRE AND A.C. MURZA We say that the meridian M(ai ,bi ) with a2i + b2i = 1/2 has multiplicity k if both invariant hyperplanes x3 − ai = 0 and x4 − bi = 0 of the differential system (1) with the polynomials Pi given by (4) have mul- tiplicities k1 and k2 respectively, and min{k1 , k2 } = k. In a similar way is defined the multiplicity of a parallel P(ai ,bi ) . Proof of Theorem 2. A meridian of the Clifford torus T is obtained by intersecting T with the hyperplanes x3 = a and x4 = b with a2 + b2 = 1/2. So the hyperplanes x3 − a = 0 and x4 − b = 0 must be invariant by the polynomial vector field X . In other words x3 − a must divide the extactic polynomial 1 x3 E1,x3 (X ) = = P3 , 0 P3 i.e. x3 − a must divide the polynomial P3 (x1 , x2 , x3 , x4 ). In a similar way x4 − b must divide the polynomial P4 (x1 , x2 , x3 , x4 ). Since the degrees of Pi is mi for i = 3, 4, it follows that the polynomials x3 − ai at most divide m3 times the polynomial P3 . This is only possible if m3 ∏ m4 ∏ (8) P3 = k3 (x3 − ai ), and P4 = k4 (x4 − bi ), i=1 i=1 with k3 , k4 ∈ R \ {0}. But taking into account the expressions of P3 and P4 given in (4) we only can obtain the expressions of (8) if and only if (9) D = 2kx3 , E = −2kx4 , and F = kx3 x4 , with k ∈ R \ {0}. Then P3 = 2kx3 (x23 + x24 − 1/2) − 2kx3 x24 = 2kx3 (x23 − 1/2), P4 = −2kx4 (x23 + x24 − 1/2) + 2kx23 x4 = 2kx4 (x24 − 1/2). So in this case m3 = m4 = 3 and we have 4 meridians, namely M(0,1/√2) , M(0,−1/√2) , M(1/√2,0) and M(−1/√2,0) . Expect this exceptional case (9) that we can obtain for the polyno- mials P3 and P4 the expressions (8), we have that there are at most m3 −2 invariant hyperplanes of the form x3 −bi taking into account their multiplicities, and at most we have m4 − 2 invariant hyperplanes of the form x4 − bi taking into account their multiplicities. These invariants hyperplanes are obtained when 3 −2 m∏ 4 −2 m∏ D = k3 (x3 − ai ), E = k4 (x4 − bi ), F = 0, i=1 i1 POLYNOMIAL VECTOR FIELDS ON THE CLIFFORD TORUS 7 with k3 , k4 ∈ R \ {0}. Therefore the differential system (1) with the polynomials Pi given by (4) has the invariant meridians M(ai ,bi ) if a2i + b2i = 1/2 for i = 1, . . . , m4 − 2. Eventually some of these invariant meridians can have multiplicity larger than one if the ai and bi appears repeated in the expressions of the polynomials D and E. In short, if m4 > 3 an upper bound for the maximum number of invariant meridians is m4 − 2 taking into account their multiplicities, because m3 ≥ m4 ; or if m4 = 3 that upper bound is 4. Note from this proof that these upper bounds are reached. This proves statement (a) of Theorem 2. In an analogous way we obtain that the maximum number of invari- ant parallels is m2 −2 taking into account their multiplicities if m2 > 3, or 4 if m2 = 3. This proves statement (b) of Theorem 2.  Acknowledgements The first author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant 2014SGR-568. The second author acknowledges support from a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4- 0657. 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