This is a preprint of: “Invariant algebraic surfaces of polynomial vector fields in dimension three”, Niclas Kruff, Jaume Llibre, Chara Pantazi, Sebastian Walcher, J. Dyn. Diff. Equat., 2021. DOI: [10.1007/s10884-021-10080-8] INVARIANT ALGEBRAIC SURFACES OF POLYNOMIAL VECTOR FIELDS IN DIMENSION THREE NICLAS KRUFF1 , JAUME LLIBRE2 , CHARA PANTAZI3 AND SEBASTIAN WALCHER4 Abstract. We discuss criteria for the nonexistence, existence and computation of invariant alge- braic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi–invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi–invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields. AMS classification (2010): 34A05, 34C45, 37F75, 17A60. Key words: Poincaré problem, Darboux integrability, Jacobi multiplier, nonassociative alge- bras. 1. Introduction Consider a polynomial ordinary differential equation in Cn ẋ = f (x) = f (0) (x) + f (1) (x) + · · · + f (m) (x), (1) with each f (i) a homogeneous polynomial of degree i, 0 ≤ i ≤ m, and f (m) 6= 0. By Xf we denote the vector field associated to f (also called the Lie derivative with respect to f ). A polynomial ψ : Cn → Cn is called a semi–invariant of f if ψ is nonconstant and Xf (ψ) = λ · ψ, (2) for some polynomial λ, called the cofactor of ψ. As it is well–known, a polynomial is a semi– invariant of the vector field f if and only if its vanishing set is invariant for the flow of system (1). Given a degree bound, the problem of finding semi–invariants essentially reduces to solving a linear system of equations with parameters, thus a problem of linear algebra. The existence problem for semi–invariants is relevant for several questions, notably Darboux integrability and existence of Jacobi multipliers. From the work of Żola̧dek [25] it is known that generically no algebraic invariant sets exist for polynomial vector fields of a fixed degree; see also Coutinho and Pereira [7]. Even for dimension n = 2 the existence problem is hard when the vector field has dicritical stationary points. When there are no dicritical stationary points then work of Cerveau and Lins Neto [5] and Carnicer [3] provides degree bounds for irreducible semi–invariants; see Pereira [15] for a refinement and also note the recent work by Ferragut et al. [9]. For certain classes of planar polynomial vector fields it was shown in [24] by elementary arguments that an effective degree bound for irreducible semi–invariants exists, and strong restrictions were found for possible integrating factors. For higher dimensions Jouanolou [10] showed the existence of the general degree bound m + 1 for semi–invariants of system (1) that define smooth hypersurfaces in projective space, and Soares [21, 22, 23] extended and refined this result. For dimension n ≥ 3 1 much work has been done in the last decade to classify and characterize invariant surfaces; see for instance the survey [4] by Cerveau, and the work [6] by Cerveau et al. on local properties. The purpose of the present paper, which is based in part on the doctoral thesis [11] by one of the authors, is to generalize the results of [24] to higher dimensions, with a focus on dimension three. In contrast to the deep theoretical results used in most of the above mentioned references, our approach is different, employing rather elementary methods. Thus we start with asking about the existence of irreducible, pairwise relatively prime semi–invariants (1) (ri ) φi = φi + · · · + φi , 1 ≤ i ≤ s, (3) (k) (r ) with φi homogeneous polynomials of degree k, and φi i 6= 0. Our principal approach will be to consider stationary points at infinity of system (1), and we obtain joint degree bounds for the φi . The plan of the paper is as follows. In a preparatory section 2 we collect mostly known facts about semi–invariants of polynomial and formal vector fields, and Poincaré transforms in order to discuss the behavior at infinity. Some of the statements are proven for the reader’s convenience. In section 3 we consider a class of polynomial vector fields in Cn which is characterized by certain properties of its stationary points at infinity. We derive degree bounds for collections of irreducible semi–invariants given that either at least n−1 pairwise relatively prime semi–invariants exist or that a degree bound for (and a bound for the number of) the irreducible homogeneous semi–invariants of the highest degree term f (m) is known. We then proceed to discuss posssible exponents and degree bounds for Jacobi multipliers that are algebraic over the rational function field C(x1 , . . . , xn ). We close the section by stating some facts about reduction of dimension for homogeneous vector fields. In section 4 we apply and specialize the results from the previous section to vector fields in C3 , obtaining rather strong results on degree bounds by combining our approach with earlier results by Jouanolou [10] and Carnicer [3]. Moreover we explicitly construct a class of quadratic vector fields for which the conditions on the stationary points at infinity are directly verifiable. This class, seen as a subset of the coefficient space (∼ = C18 ) contains a Zariski open subset, and the vector fields not satisfying the conditions on stationary points at infinity form a measure zero subset. The Appendix contains some additional material on the construction and some proofs. It also contains the statement and proof of a result by Röhrl [18] (the original source contains an erroneous statement and proof) which is needed by us as a basis for the construction of the quadratic vector fields in section 4. Röhrl’s result, which generalizes the common knowledge fact that generically a linear map admits a basis of eigenvectors to homogeneous polynomial maps of arbitrary positive degree, seems to be of independent interest. 2. Preparations In this section we review some known facts and tools to be used later on. 2.1. Semi–invariants and some of their properties. In addition to polynomial semi–invariants of polynomial vector fields we will consider local analytic (or formal) semi–invariants of local analytic (or formal) vector fields. Thus, a formal vector field is given by a power series X g(x) = Bx + g (i) (x) in Cn , (4) i≥1 with B linear and each g (i) a homogeneous polynomial of degree i. A semi–invariant of g is defined as a non-invertible power series ρ = ρ(1) + ρ(2) + · · · (thus ρ(0) = 0) (5) 2 satisfying Xg (ρ) = µρ for some power series µ. Similar to the polynomial case, an analytic function at 0 is a semi–invariant of the analytic vector field (4) if and only if its vanishing set is invariant for ẋ = g(x). We collect some general properties of semi–invariants. Lemma 1. (a) Let the polynomial vector field f as in (1) be given. Then the following hold. • From Xf (ψj ) = λj · ψj , j = 1, 2, the relations Xf (ψ1 · ψ2 ) = (λ1 + λ2 ) · ψ1 · ψ2 and Xf (ψ1 /ψ2 ) = (λ1 − λ2 ) · ψ1 /ψ2 follow. • If ψ1 , . . . , ψ` are irreducible and pairwise relatively prime polynomials, m1 , . . . , m` are nonzero integers and there is a polynomial µ such that Xf (ψ1m1 · · · ψ`m` ) = µ · ψ1m1 · · · ψ`m` then every ψj is a semi-invariant of f . • If σ is nonconstant and algebraic over the field C(x1 , . . . , xn ) and satisfies Xf (σ) = µ · σ for some polynomial µ then every nonzero coefficient β of its minimal polynomial satisfies Xf (β) = k · µ · β with some positive integer k, hence also Xf (β 1/k ) = µ · β 1/k . (b) Mutatis mutandis, the same statements hold for formal semi-invariants of formal vector fields. Proof. The proofs for statement (b) are parallel to those for statement (a); so we only consider these. The first statement is straightforward, while the second is a direct consequence of the derivation property of Xf and unique factorization in the polynomial ring. For the third statement, let T m + β1 T m−1 + · · · + βm ∈ C(x1 , · · · , xn )[T ] be the minimal polynomial of σ. Applying Xf to σ m +β1 σ m−1 +· · ·+βm = 0 and using Xf (σ) = µ·σ, one gets 0 = µ · (mσ m + β1 (m − 1)σ m−1 + · · · + βm−1 σ) + Xf (β1 )σ m−1 + · · · + Xf (βm−1 )σ + Xf (βm ). Subtract this relation from m · µ · (σ m + β1 σ m−1 + · · · + βm ) = 0 to obtain 0 = (µ · β1 − Xf (β1 ))σ m−1 + · · · + (m · µ · βm − Xf (βm )). Since the minimal polynomial of σ has degree m, all coefficients must vanish. Hence, for nonzero βk one finds Xf (βk ) = k · µ · βk .  Given a semi–invariant ψ of the polynomial vector field f , and a stationary point z of f , one has either ψ(z) 6= 0 or ψ is a local analytic (hence also a formal) semi–invariant of f at z. Thus one can use local information in the search for polynomial semi–invariants, based on the following result. Lemma 2. The following statements hold. (a) Let ψ be an irreducible polynomial semi-invariant of (1), and ψ(0) = 0. Then every irreducible series in the factorization of ψ in the formal power series ring C[[x1 , . . . , xn ]] has multiplicity one. (b) Let ψ1 and ψ2 be relatively prime polynomial semi-invariants of (1), and ψ1 (0) = ψ2 (0) = 0. Then the prime factorizations of ψ1 and ψ2 in the formal power series ring C[[x1 , . . . , xn ]] have no common irreducible factor. These properties hold more generally for polynomials. An algebraic proof of this fact (which should be considered as known) is given in [11], Lemma 8.4, and we give an elementary ad-hoc proof in subsection 5.1 of the Appendix. We next discuss cases when the local information is rather precise. Lemma 3. Let g be a formal vector field as in (4), with λ1 , · · · , λn the eigenvalues of B. Consider the following conditions: 3 1. λ1 , · · · , λn are linearly independent over the rational number field Q. 2. dimQ (Qλ1 + · · · + Qλn ) = n − 1 and there exist positive integers m1 , . . . , mn (w.l.o.g. relatively prime) with X n mi λi = 0. i=1 If one of the two conditions above is satisfied then the following hold. (a) If B = diag(λ1 , . . . , λn ) and g is in Poincaré–Dulac normal form (PDNF) then x1 , . . . , xn are (up to multiplication by invertible series) the only irreducible formal semi–invariants of g. (b) For general g there exist (up to multiplication by invertible series) precisely n irreducible and pairwise relatively prime formal semi–invariants of system (4). Proof. It suffices to prove statement (a), since a formal transformation to PDNF always exists. Thus let g be in PDNF and let ρ be an irreducible semi–invariant. According to Lemma 2.2 of [24], up to multiplication with an invertible series one may assume that XB (ρ) = αρ for some constant α, and Xg (ρ) = µρ with XB (µ) = 0. If condition 1 holds then one has g(x) = Bx (see e.g. Bibikov [2], Theorem 2.1), and for the series expansion X ρ= cd1 ,...,dn xd11 · · · xdnn one obtains ! X X αρ = XB (ρ) = di λi cd1 ,...,dn xd11 · · · xdnn , i thus X α= di λi whenever cd1 ,...,dn 6= 0. (6) Since the λi are linearly independent over Q, one sees that only one coefficient cu1 ,...,un is nonzero, and therefore ρ = xu1 1 · · · xunn . The assertion about irreducible factors follows. If condition 2 holds then one also arrives at (6), but now, given distinct (d1 , · · · , dn ) and (e1 , · · · , en ) with nonnegative integer entries such that X X α= di λi = ei λi , one has that di −ei = `·mi with some rational number `, due to the dimension assumption. Since the mi are relatively prime one sees that ` is an integer. Define γ(x) := xm 1 mn 1 · · · xn , noting P XB (γ) = 0. Let P (u1 , · · · , un ) be a nonzero vector with nonnegative integer entries such that m · i ui = 0, and ui minimal with respect to these properties. Then, repeated use of the argument above shows the existence of a nonnegative integer j such that xd11 · · · xdnn = γ(x)j · xu1 1 · · · xunn whenever cd1 ,...,dn 6= 0. This shows X ρ = xu1 1 · · · xunn · cj γ(x)j b and the only irreducible factors of ρ are the xi .  Thus, one only has to consider finitely many local semi–invariants if condition 1 or 2 from Lemma 3 holds. Moreover, their vanishing sets (in the analytic case) meet transversally in the stationary point 0. This property is characteristic for conditions 1 and 2 above, as the next observation shows. Remark 1. When the dimension is n ≥ 3 and dimQ (Qλ1 + · · · + Qλn ) ≤ n − 2, then there always exist infinitely many irreducible, pairwise relatively prime formal semi–invariants P for the semisimple part Bs of B: By the dimension assumption, there is a relation `i λi = 0 with integers `i that 4 do not all have the same sign; thus we may assume that there are nonnegative, relatively prime integers ki and some 1 < p < n such that p X n X ki λi = ki λi , i=1 i=p+1 k k whence xk11 · · · xpp and xp+1 p+1 · · · xknn are semi-invariants with the same cofactor, and k k xk11 · · · xpp + α · xp+1 p+1 · · · xknn is a semi-invariant for every constant α. Since the ki are relatively prime, irreducibility follows whenever α 6= 0. 2.2. Poincaré transforms and stationary points at infinity. The following is geometrically motivated by the well known procedure of passing to the Poincaré hypersphere (or to projective space) in the analysis of polynomial vector fields. We choose a rather straightforward adaptation (see [24] in the case of dimension two), to facilitate our computations. Definition 1. Let ψ ∈ C[x] be a polynomial of degree r with decomposition r X ψ(x) = ψ (j) (x), j=0 where each ψ (j) is a homogeneous polynomial of degree j and ψ (r) 6= 0. (a) The homogenization of ψ with respect to xn+1 is defined as r X e 1 , . . . , xn , xn+1 ) := ψ (j) (x1 , . . . , xn )xr−j ψ(x n+1 ∈ C[x1 , . . . , xn , xn+1 ]. j=0 (b) The special Poincaré transform of ψ with respect to e1 = (1, 0, . . . , 0)T ∈ Cn is the polynomial e x2 , . . . , xn+1 ) ψ ∗ := ψe∗1 (x2 , . . . , xn+1 ) :=ψ(1, r X = ψ (j) (1, x2 , . . . , xn )xr−j n+1 ∈ C[x2 , . . . , xn , xn+1 ]. j=0 (c) A Poincaré transform of ψ w.r.t. v ∈ Cn \ {0} is defined as ∗ ψv∗ (x2 , . . . , xn+1 ) := ψ ◦ T −1 e1 (x2 , . . . , xn+1 ) with a regular matrix T ∈ Cn×n such that T v = e1 . The following properties are easy to prove, see [24] or [11] for more details. Lemma 4. Let ψ and v be as in Definition 1. Then the following hold. (a) One has ψ (r) (v) = 0 if and only if ψv∗ (0) = 0. (b) The map C[x1 , · · · , xn ] \ hx1 i → C[x2 , · · · , xn+1 ], ψ 7→ ψe∗1 is injective. (c) If ψ is irreducible with ψ (r) (v) = 0 then ψv∗ is irreducible. (d) If ψ1 and ψ2 are relatively prime and ψ1∗ (v) = ψ2∗ (v) = 0, then ψ1∗ and ψ2∗ are relatively prime. 5 Moreover, we define Poincaré transforms of vector fields. Definition 2. Let f be given as in (1). (a) The homogenization of f with respect to xn+1 is defined as   m  g1   P (j) m−j  .  x1 f (x1 , . . . , xn )xn+1 . fe(x1 , . . . , xn+1 ) := j=0  =:   .   .  ∈ C[x]n+1 , where x =  ..  . gn  0 xn+1 0 (b) The projection of fe with respect to x1 is P fe(x1 , . . . , xn+1 ) := − g1 (x1 , . . . , xn+1 ) · x + x1 · fe(x1 , . . . , xn+1 )   0  −g1 x2 + x1 g2     ..  = . .   −g1 xn + x1 gn  −g1 xn+1 (c) The special Poincaré transform with respect to the vector e1 is defined as   −g1 (1, x2 , . . . , xn+1 )x2 + g2 (1, x2 , . . . , xn+1 )  ..   .  f ∗ := fe∗1 (x2 , . . . , xn , xn+1 ) :=  , −g1 (1, x2 , . . . , xn+1 )xn + gn (1, x2 , . . . , xn+1 ) −g1 (1, x2 , . . . , xn+1 )xn+1 where f ∗ ∈ C[x2 , . . . , xn , xn+1 ]n . (d) A Poincaré transform of f w.r.t. v ∈ Cn \ {0} is defined as ∗ fv∗ (x2 , . . . , xn+1 ) := T ◦ f ◦ T −1 e1 (x2 , . . . , xn+1 ) with a regular matrix T ∈ Cn×n such that T v = e1 . Remark 2. The definitions of Poincaré transforms depend on the choice of the matrix T , but a different choice of T will just amount to a linear automorphism of the polynomial algebra resp. a conjugation of vector fields by a linear transformation; see [24], Lemma 3.1. This will be irrelevant for our purpose. We now turn to stationary points at infinity, i.e. to stationary points of Poincaré transforms which lie in the hyperplane {x : xn+1 = 0}. Lemma 5. Let f be as in (1) and v ∈ Cn \ {0}. Then: (a) The point 0 is stationary for fv∗ if and only if f (m) (v) = γv for some γ ∈ C. (b) The Jacobian Df (m) (v) admits the eigenvector v, with eigenvalue mγ. Let β2 , . . . , βn be the further eigenvalues of Df (m) (v) (counted according to multiplicity). Then the eigenvalues of Dfv∗ (0) are given by −γ, β2 − γ, . . . , βn − γ. (c) If the number of lines Cv with f (v) ∈ Cv is finite then it is equal to (m) n−1 mn − 1 X i = m, m−1 i=0 counting multiplicities. 6 Proof. Statement (a) can be verified directly. For statement (b) see [19], Proposition 1.8 and Corollary. 1.9. For statement (c), a proof is given by Röhrl [17], using Bezout’s theorem in projective space (see e.g. Shafarevich [20], Chapter 4 for this).  Remark 3. Note that the stationary point 0 of fv∗ has multiplicity one if and only if all the eigenvalues of its Jacobian are nonzero. Moreover, the stationary point 0 is then isolated. In particular, given condition 1 or 2 of Lemma 3 for the eigenvalues at the stationary point 0 of fv∗ , this stationary point has multiplicity one. 3. Invariant hypersurfaces of a class of vector fields In the present section we will discuss a particular class of vector fields and degree bounds for irreducible semi–invariants of this class. We first give some definitions. Definition 3. Let f be given as in (1). (a) We say that f has property E if condition 1 or condition 2 in Lemma 3 holds for the linearization of fv∗ at every stationary point at infinity. (b) We say that f has property S if every proper homogeneous invariant variety Y of f (m) with dim Y ≥ 1 contains an invariant subspace Cv with v 6= 0. Note that both conditions apply only to the highest degree term f (m) . In dimension two, prop- erty S is trivially satisfied, and it was shown in [24], Proposition 3.7 that property E is generic. In higher dimensions it is not a priori obvious that vector fields admitting properties E and S exist. In section 4 we will construct a class of examples in dimension three. One consequence of property E is that the number of stationary points at infinity is finite (oth- erwise some Jacobian at a stationary point would have to be non-invertible), and every stationary point at infinity has multiplicity one. A further consequence is not directly relevant for the main topic of this paper, but it is worth recording. Proposition 1. Let f satisfy property E. Then the number of irreducible invariant algebraic curves for system (1) is bounded by (mn − 1)/(m − 1). Proof. Let C be an invariant algebraic curve of the system. Then its intersection with the hyper- plane at infinity is invariant, hence every intersection point is stationary. Let Cv correspond to one of these points. Then, by Theorems 3.1 and 3.2 of [12], the local ideal defining the image of the curve under the Poincaré transform is generated by certain semi-invariants of Dfv∗ (0), and there must be n−1 of these, since the dimension of the curve equals one. Using property E and Lemma 3, and the fact that the transformed curve is not contained in the hyperplane {x : xn+1 = 0}, one sees that the only possible ideal is the one not containing the semi–invariant xn+1 . This argument shows that every stationary point at infinity can be an intersection point with at most one irreducible invariant curve. Now use Lemma 5(c).  3.1. Degree bounds for semi–invariants. We first state a preliminary result. Lemma 6. Let φ1 , . . . , φd be irreducible and pairwise relatively prime semi–invariants of the poly- nomial vector field f , and let v be such that φ∗i,v (0) = 0, 1 ≤ i ≤ d, and that the Jacobian Dfv∗ (0) satisfies condition 1 or 2 from Lemma 3. Denote the irreducible local semi–invariants of Dfv∗ (0) by σ1 , . . . , σn−1 , σn := xn+1 . Then σn is not a factor of any φ∗i,v , and for the prime decompositions ` ` φ∗i,v = σ1i,1 · · · σn−1 i,n−1 · νi , 1≤i≤d 7 with nonnegative integers `i,j and invertible series νi one has X `i,j ≤ 1. i Proof. This is a direct consequence of Lemma 2.  Our first result yields degree bounds when n − 1 irreducible semi–invariants exist. Theorem 1. Let the vector field f in (1) satisfy properties E and S, and let φ1 , . . . , φn−1 be irreducible and pairwise relatively prime semi–invariants of f as in (2), of degrees r1 , . . . , rn−1 respectively. Then n−1 Y mn − 1 ri ≤ . m−1 i=1 Proof. We first recall that invariance of the common zero set of the φj for (1) implies invariance (r ) (rn−1 ) (r ) (rn−1 ) of V(φ1 1 , . . . , φn−1 ) for ẋ = f (m) (x). Now let Cv be a common zero of φ1 1 , . . . , φn−1 . By property S we may assume that f (m) (v) ∈ Cv. Consider the prime factorization of the φ∗i,v in the power series ring. By property E and Lemma 6, after possibly renumbering the factors, we have φ∗i,v = σi · νi , 1 ≤ i ≤ d, whence the intersection multiplicity of the common zero of the φ∗i,v and xn+1 is equal to one. In (r ) (r ) particular the irreducible component Y of V(φ1 1 , . . . , φn−1 n−1 ) that contains Cv is equal to Cv. (r ) (r ) Bezout’s Theorem in projective space now shows that V(φ1 1 , . . . , φn−1 n−1 ) is the union of precisely Qn−1 r i=1 i distinct lines in C n . Each of these lines is an invariant set for ẋ = f (m) (x), hence by Lemma 5(c) one has the assertion.  Corollary 1. Let f be as in (1), satisfying properties E and S. Moreover, let k ≥ n and φ1 , . . . , φk be irreducible and pairwise relatively prime semi–invariants of f , with deg φi = ri . Then, Y X 1 mn − 1 ri · Q ≤ . j∈M rj m−1 1≤i≤k M ⊆ {1, . . . , k} |M | = k + 1 − n Proof. Let 1 ≤ i1 < · · · < in−1 ≤ k, thus {i1 , . . . , in−1 } is the complement of a set M ⊆ {1, . . . , k} with k + 1 − n elements. Then, by the argument in the proof of Theorem 1, the vanishing set  (r ) (rin−1 )  i V φi1 1 , . . . , φin−1 is a union of precisely ri1 · · · rin−1 invariant lines, each with multiplicity one. By property E and Lemma 3(b), each invariant line Cv of f (m) appears in at most one of the common vanishing sets  (r ) (rin−1 )  i V φi1 1 , . . . , φin−1 , since otherwise more than n local invariant hypersurfaces would meet at the stationary point 0 of fv∗ . Now Bezout’s theorem, and adding up the contributions of all stationary points at infinity, shows the assertion.  The second result may be used to obtain degree bounds for semi–invariants of f , if degree bounds for semi–invariants of the highest degree term f (m) are known. Theorem 2. Let the vector field f in (1) satisfy properties E and S, and let φ1 , . . . , φs be irreducible and pairwise relatively prime semi–invariants of f . 8 (r ) (a) Let {ψ1 , . . . , ψ` } be the set of pairwise relatively prime irreducible factors of the φi i with 1 ≤ i ≤ s. (Note that these are homogeneous and semi–invariants of f (m) .) Then, given the representations (r ) Ỳ kij φi i = ψj , kij ≥ 0, j=1 Ps (r ) one has i=1 kij ≤ 1 for all j. Thus every ψj appears in at most one of the φi i , and if it appears then with exponent 1. (b) If f (m) admits only finitely many irreducible semi–invariants ψ1 , . . . , ψ` (up to multiplication by constants), then X̀ deg φi = kij deg ψj , 1≤i≤s j=1 and s X X̀ deg φi ≤ deg ψj . i=1 j=1 In particular the number of irreducible and pairwise relatively prime semi–invariants of f is finite. Proof. Statement (b) is a simple consequence of (a). As for the proof of statement (a), by property (r ) S we may again consider the φi i at a common zero Cv which is also invariant for ẋ = f (m) (x). We consider a Poincaré transform at v, and there is no loss of generality in assuming that v = e1 . Moreover let the σi be as in Lemma 6, and by property E and Lemma 3 we may assume that σi (x2 , . . . , xn , xn+1 ) = xi+1 + h.o.t, 1 ≤ i ≤ n − 1, with “h.o.t” denoting higher order terms. From Definition 1 we have (ri ) ` ` φi (1, x2 , . . . , xn ) + xn+1 · (· · · ) = φ∗i,e1 = σ1i,1 · · · σn−1 i,n−1 · νi with all `i,j ∈ {0, 1} and therefore also (ri ) φi (1, x2 , . . . , xn ) = σ1 (x2 , . . . , xn , 0)`i,1 · · · σn−1 (x2 , . . . , xn , 0)`i,n−1 · νi (x2 , . . . , xn , 0). From σi = xi+1 + · · · one sees that the σi (x2 , . . . , xn , 0) are still irreducible and pairwise relatively prime in the formal power series ring. Comparing this decomposition with (ri ) Ỳ k φi = ψj ij , j=1 k the corresponding local decompositions of the ψj ij (1, x2 , . . . , xn ) can have only simple prime factors, hence kij ≤ 1 whenever ψj (v) = 0.  3.2. Degree bounds for Jacobi multipliers. We recall that a Jacobi multiplier (or Jacobi last multiplier) σ 6= 0 of a vector field h is characterized by the condition Xh (σ) + div h · σ = 0, equivalently Xh (σ −1 ) = div h · σ −1 . In particular, a Jacobi multiplier satisfies the defining identity for semi–invariants but might be invertible. For a comprehensive account of Jacobi multipliers and their properties see Berrone and Giacomini [1]. We focus on Jacobi multipliers that are algebraic over C(x1 , . . . , xn ) resp. C((x1 , . . . , xn )). This is a natural requirement in dimension n = 2 in view of Prelle and Singer’s paper [16] on elementary 9 integrability, and seems to be a sensible ( initial) restriction also for local integrating actors of Darboux type, as the following results will imply. We note that by Lemma 1 one may restrict interest to products of powers of semi–invariants, with rational exponents. We first discuss the local analytic, respectively the formal case, given the setting of Lemma 3. Lemma 7. Let g(x) = Bx + · · · as in (4) in Poincaré-Dulac normal form, B = diag (λ1 , . . . , λn ). (a) If condition 1 from Lemma 3 holds, then (x1 · · · xn )−1 is, up to multiplication by constants, the only Jacobi multiplier of g which is algebraic over C((x1 , . . . , xn )). (b) If condition 2 from Lemma 3 holds, and γ = xm mn 1 · · · xn , then there exist linearly indepen- 1 dent diagonal matrices C1 , . . . , Cn−1 such that the Lie bracket [g, Ci x] = 0 and XCi x (γ) = 0, 1 ≤ i ≤ n − 1; moreover B is a linear combination of the Ci . – If τ (x) := det(g(x), C1 x, . . . , Cn−1 x) 6= 0, then there exists ` > 0 such that τ (x) = γ(x)` · (x1 · · · xn ) · τb(x), with an invertible series τb, and τ −1 is (up to multiplication by constants) the unique Jacobi multiplier for g which is algebraic over C((x1 , . . . , xn )). – If τ (x) = 0 then (x1 · · · xn )−1 is a Jacobi multiplier, and γ a first integral of g. In that case every Jacobi multiplier that is algebraic over C((x1 , . . . , xn )) has the form (x1 · · · xn )−1 · ν with d ∈ Q and ν an algebraic first integral of g. Proof. (a) In this case one has g(x) = Bx, and one directly verifies that σ := (x1 · · · xn )−1 is a Jacobi multiplier. Assume that there exists a Jacobi multiplier σ e that is algebraic over C((x1 , . . . , xn )), −d1 −d then one also has a Jacobi multiplier x1 · · · xn with rational exponents by Lemma 1, and n X X di λi = λi . Condition 1 implies that all di = 1, hence x−1 −1 1 · · · xn is the unique algebraic integrating factor. Using the (notation and) argument in the proof of Lemma 1 the coefficient of T m−i in the minimal polynomial of σe−1 is a constant multiple of x1 · · · xn . But then the same holds for σ e−1 itself. (b) It is known that X g(x) = Bx + γ(x)j Dj x j≥1 with diagonal matrices Dj , see e.g. Bibikov [2], Definition 2.3 and Theorem 2.2. The equation P mi µi = 0 has n − 1 linearly independent solutions in Qn . Take these as the diagonal elements of C1 , · · · , Cn−1 respectively. Then XP Ci (γ) = 0 and [Ci , B] = [Ci , Dj ] = 0 for all i, j, whence [Ci , g] = 0 for all i. Now write Dj x = νjh Ch x + βj · x with constants νjh and βj . Thus, X X τ (x) = γ(x)j βj det(x, C1 x, . . . , Cn−1 x) = κx1 · · · xn βj γ(x)j , j≥1 j≥1 with a nonzero constant κ. If τ 6= 0 then τ −1 is a Jacobi multiplier according to Berrone and Giacomini [1], and if ` is the smallest index with β` 6= 0 then we get τ (x) = γ ` (x) · (x1 · · · xn ) · (β` κ + h.o.t). As for uniqueness, we first recall: Every first integral in C((x1 , . . . , xn )) of ẋ = Bx is a quotient of power series in γ. Indeed, numerator and denominator are semi-invariants of B, with the same cofactor (which is a first integral of B and lies in C[[x1 , . . . , xn ]]), and the argument in the proof of Lemma 3 shows the assertion. From this we find that g admits no nonconstant first integral that is algebraic over C((x1 , . . . , xn )) whenever some β` 6= 0, by showing that there exists no such first integral in C((x1 , . . . , xn )). But 10 such a first integral would also be a first integral of ẋ = Bx (see e.g. [14], Theorem 1), hence a quotient of power series in γ. But then γ is a first integral. Finally, the identity X X Xg (γ) = κ · mi · βj γ(x)j+1 j≥1 yields a contradiction unless all βj = 0. In the case τ = 0 one verifies by direct computation that (x1 · · · xn )−1 is a Jacobi multiplier, and γ is obviously a first integral of g. The last statement is again clear from known properties of Jacobi multipliers; see [1].  Obviously, Lemma 7 is also applicable to local analytic or formal systems which are not in PDNF, given that condition 1 or condition 2 of Lemma 3 holds. We shall now apply this to Jacobi multipliers of system (1) that are algebraic over the rational function field C(x1 , . . . , xn ). Theorem 3. Let the polynomial vector field f given by (1) satisfiy properties E and S and let  −1 φd11 · · · φds s be a Jacobi multiplier of f , with the φi as in (3), and nonzero rational exponents d1 , · · · , ds . Moreover, assume that there exists a line Cw such that linearization of fw∗ at 0 has eigenvalues that are linearly independent over Q. Then Xs d1 = · · · = ds = 1 and ri = m + n − 1. i=1 Proof. (i) We need some technical preparations, the proofs of which are straightforward general- izations of the ones for Proposition 3.3 in [24]. • If ψ is a semi–invariant of f , with degree r and Xf (ψ) = λψ, then the Poincaré transform ψe∗1 is a semi–invariant of fe∗1 with cofactor −rg1 (1, x2 , . . . , xn+1 ) + λ∗e1 . • fe∗1 admits the Jacobi multiplier  P −1 (m+n− di ri ) xn+1 · (φ∗1,e1 )d1 · · · (φ∗s,e1 )ds . • More generally, for any v ∈ Cn \{0}, the Poincaré transform fv∗ admits the Jacobi multiplier  P −1 (m+n− di ri ) xn+1 · (φ∗1,v )d1 · · · (φ∗s,v )ds . (ii) Let Cw correspond to a stationary point at infinity such that the eigenvalues of Dfw∗ (0) are linearly independent over Q. Then Lemma 7 (a) shows that di = 1 for all i such that φ∗i,w (0) = 0, P and moreover m + n − di ri = 1. (iii) If Cv corresponds to a stationary point at infinity such that the eigenvalues of Dfv∗ (0) satisfy the second condition of Lemma 3, then one exponent in the local factorization of the multiplier, viz. the one belonging to the factor xn+1 , is equal to −1. By Lemma 7(b), all the other exponents equal −1, hence di = 1 whenever φ∗i,v (0) = 0. Due to property S, we thus find that all exponents are equal to −1, and the assertion follows.  3.3. Reduction of dimension. The verification of property S in dimension three, as well as the search for degree bounds via Theorem 2 leads to semi–invariants of the homogeneous vector field f (m) , and in turn to semi–invariants of a reduced vector field in Cn−1 . This reduction is due to scaling symmetry, and we will recall it now. Proposition 2. Let p : Cn → Cn , x 7→ (p1 (x), . . . , pn (x))T be homogeneous of degree m, and let H = {x : xn = 0} . 11 (a) Then   x1 /xn  ..  Φ : Cn \ H → Cn−1 , x 7→  .  xn−1 /xn maps solutions orbits of ẋ = p(x) to solution orbits of ẏ = q(y), with   q1   q =  ...  ; qi (y) = pi (y1 , · · · , yn−1 , 1) − yi pn (y1 , · · · , yn−1 , 1). qn−1 (b) Every homogeneous invariant set Y of ẋ = p(x) which is not contained in H is mapped to an invariant set of ẏ = q(y) with dimension decreasing by one. Conversely, the inverse image of every invariant set Z of ẏ = q(y) is a homogeneous invariant set of ẋ = p(x), with dimension increasing by one. (c) Let v ∈ Cn such that vn 6= 0 and p(v) = γv. Then v is an eigenvector of Dp(v) with eigenvalue mγ. Let β2 , . . . , βn be the other eigenvalues of Dp(v), each counted according to its multiplicity. Then the linearization of q at the stationary point (v1 /vn , . . . , vn−1 /vn ) has eigenvalues vn (β2 − γ), . . . , vn (βn − γ). Sketch of proof. From ẋi = pi (x) one gets   d xi 1 = (xn pi (x) − xi pn (x)), 1≤i≤n−1 dt xn xn and rescaling time yields  0 xi = xn pi (x) − xi pn (x), 1 ≤ i ≤ n − 1. xn Dehomogenize to obtain statement (a). Statement (b) is a direct consequence of (a) since any invariant set is a union of solution orbits. The proof of statement (c) is a variant of the proof of [19], Proposition 1.8: Define Q(x) := xn · p(x) − pn (x) · x, with DQ(x)y = yn · p(x) + xn · Dp(x)y − (Dp(x)y)n · x − pn (x)y and note that the first n − 1 entries of Q(x), upon setting xn = 1, are just the entries of q(x). Now let y be an eigenvector of Dp(v) that is linearly independent from v, with eigenvalue β. Then, using p(v) = γv, one gets DQ(v)y = (· · · ) · v + vn · Dp(v)y − γvn · y = (· · · ) · v + vn · (β − γ) · y. This proves the part of the assertion for eigenvectors. The remaining part (if nontrivial Jordan blocks exist) is proven similarly.  Remark 4. (a) Note that the entries of q are just the first n − 1 entries of the Poincaré transform of p with respect to en . (b) A coordinate–free version of the reduction starts from a nonzero linear form X 1 α(x) = αi xi , Hα := {x : α(x) = 0} and Ψα : Cn \ Hα → Cn , x 7→ x. α(x) Then Ψα maps solution orbits of ẋ = p(x) to solution orbits of the equation ẋ = Qα (x) := α(x)p(x) − α(p(x))x 12 which admits the linear first integral α. In this case, whenever p(v) = γv, α(v) 6= 0 and the eigenvalues of Dp(v) are as in Proposition 2 , the eigenvalues for the reduced system on the hyperplane given by α(x) = 1 are α(v) · (β2 − γ), . . . , α(v) · (βn − γ). 4. Dimension three In this section we will specialize our general results to dimension n = 3. In particular we will verify that property S from Definition 3 is always satisfied, and show that property E holds for almost all quadratic vector fields (in a sense to be specified). 4.1. Property S and reduction. The first pertinent property is always satisfied in dimension three, as follows directly from the work of Jouanolou [10]. Proposition 3. Let f be a polynomial vector field in C3 . Then f satisfies property S. Proof. In dimension three one has to prove that the zero set of a homogeneus semi–invariant of a homogeneous polynomial vector field p contains an invariant line for p. There is no loss of generality in assuming that the entries of p are relatively prime. One may rephrase this for the projective plane P2 (C), by introducing the one form ω = p1 dx1 + p2 dx2 + p3 dx3 and considering a homogeneous polynomial solution φ of the Pfaffian equation ω = 0. First consider the case when the projective curve defined by the zeros of φ is smooth (hence normal). Then by Jouanolou, Chapter 2, Proposition 4.1(ii), the curve defined by φ in P2 contains a singular point of ω, which corresponds to an invariant line for the homogeneous vector field p. If the normality requirement for the solution is not satisfied then the projective curve defined by φ contains a singular point, which translates to a singular line in homogeneous coordinates. But singular sets of invariant varieties of polynomial vector fields are also invariant.  We note that from Jouanolou [10], Chapter 2, Proposition 4.1(iii) one also obtains the degree bound m + 1 for irreducible homogeneous semi–invariants whose associated projective curve is smooth. Next we discuss the reduction of the highest degree term of f , with a view on applying Theorem 2. Lemma 8. Let the polynomial vector field f be given as in (1), and consider the reduction of its homogeneous highest degree term p = f (m) , according to Proposition 2 and Remark 4. Then the following hold. (a) Upon identifying homogeneous polynomial vector fields with their coefficients in some CN , the reduction q of p admits no stationary points at infinity for a Zariski–open subset of CN . (b) Assume p(v) = v and let the linear form α be such that α(v) 6= 0. Moreover let the eigenvalues of Dp(v) be m, β2 and β3 . Then the eigenvalues of the linearization of q at the corresponding stationary point are α(v)(β2 − 1) and α(v)(β3 − 1). (c) Assume that p(v) = v and the linearization of fv∗ at the stationary point at infinity of system (1) satisfies condition 1 or condition 2 from Lemma 3. Then the βi − 1 are nonzero and their ratio is not a positive rational number. Proof. We may assume that α(x) = x3 . For statement (a), abbreviate hi (y1 , y2 ) := pi (y1 , y2 , 1), noting that the hi generically have degree m. We now compute the Poincaré transform of q with 13 respect to e1 , following the procedure in Definition 2. The first two entries of the homogenization have the form   !  ! (m) (m) y1 h1 (y1 , y2 ) (m−1) y −h3 (y1 , y2 ) · + y3 · (m) − h3 (y1 , y2 ) · 1 + y32 · · · , y2 h2 (y1 , y2 ) y2 from which the Poincaré transform is computed as ! (m) (m) −h1 (1, y2 ) · y2 + h2 (1, y2 ) qe∗1 (y2 , y3 ) = y3 · (m) + y32 · · · −h3 (1, y2 ) Reducing this vector field by dividing out the factor y3 yields a vector field that generically (i.e. corresponding to a Zariski open set in the space of coefficients of the hi , hence also of the coefficients (m) (m) (m) of p) has no stationary points on y3 = 0, since −h1 (1, y2 )·y2 +h2 (1, y2 ) and h3 (1, y2 ) generically have no common zeros. Statement (b) is a direct consequence of Proposition 2. To prove statement (c), note first that conditions 1 and 2 both imply that all eigenvalues for the Poincaré transform are nonzero and observe Lemma 5. Now assume that (β3 − 1)/(β2 − 1) = r/s with positive integers r and s. Then one obtains (r − s) · (−1) + r · β2 + (−s) · β3 = 0. Therefore the eigenvalues of Dfv∗ (0) are linearly dependent over Q, hence condition 1 cannot hold. Moreover, condition 2 also cannot hold because the integer coefficients in the linear combination have different signs.  Now we are ready to determine degree bounds, applying a result of Carnicer. Theorem 4. Let the polynomial vector field f of degree m be given on C3 , and assume that (i) the reduction of the homogeneous highest degree term f (m) admits no stationary points at infinity; (ii) the vector field has property E. Then the following hold. (a) Every irreducible homogeneous semi–invariant of f (m) has degree ≤ m + 1. (b) There exist (up to scalar multiples) only finitely many irreducible homogeneous semi–invariants ψ1 . . . , ψ` of f (m) , and in case ` ≥ 2 one has X deg ψi · degψj ≤ (mn − 1)/(m − 1); 1≤i 0 and φ(x1 , . . . , xn ) = xm m−1 n + α1 (x1 , . . . , xn−1 )xn + · · · + αm (x1 , . . . , xn−1 ) ∈ C[x1 , . . . , xn ] be a polynomial with φ(0, . . . , 0) = 0. Then no prime factor of φ in the formal power series ring C[[x1 , . . . , xn ]] lies in C[[x1 , . . . , xn−1 ]]. Indeed, a factorization φ(x1 , . . . , xn ) = σ(x1 , . . . , xn ) · τ (x1 , . . . , xn−1 ) with non-invertible τ , hence τ (0, . . . , 0) = 0, would yield the contradiction φ(0, . . . , 0, xn ) = 0. 2. We now show: If φ and γ are relatively prime polynomials in n variables, with φ(0) = γ(0) = 0, then φ and γ remain relatively prime in the power series ring C[[x1 , . . . , xn ]]. (This implies statement (b) of Lemma 2.) To show this we may assume that φ is as above and likewise γ = xkn + β1 (x1 , . . . , xn−1 )xnk−1 + · · · + βk (x1 , . . . , xn−1 ) with some k > 0. (One may achieve such a form by applying a linear transformation and multiplication by nonzero constants, which does not affect the statement.) Since φ and γ are relatively prime polynomials, their resultant ρ with respect to xn lies in C[x1 , . . . , xn−1 ], and there exist polynomials µ and ν such that µ · φ + ν · γ = ρ; 17 see Cox et al. [8], Chapter 3, §6, Proposition 1. Assuming that a common prime factor of φ and γ exists in C[[x1 , . . . , xn ]], this factor also divides ρ and therefore lies in C[[x1 , . . . , xn−1 ]]; a contradiction. 3. To prove statement (a) of Lemma 2, assume that φ has the form above and is irreducible, then apply the previous argument to φ and ∂φ/∂xn to see that φ cannot admit multiple prime factors in C[[x1 , . . . , xn ]]. 5.2. A result of H. Röhrl. In [18], Röhrl stated a theorem on criteria for (in his terminology) m-ary algebras to admit a basis of idempotents. While the theorem as stated is incorrect, the weaker statement that generically an m-ary algebra admits a basis of idempotents is correct, and Röhrl’s arguments can be modified to prove it. Here we give a proof that in part takes a different approach. Consider a homogeneous polynomial map   X p : Cn → Cn ; p(x) =  ηi1 ,...,in ,k xi11 · · · xinn  (11) i1 ,...,in 1≤k≤n of degree m > 1. As noted above, an idempotent of p is a v ∈ Cn such that p(v) = v 6= 0. Note that any w 6= 0 that satisfies p(w) = β · w for some β 6= 0 is a scalar multiple of an idempotent. (If w 6= 0 and p(w) = 0 then w is a nilpotent.) In order to state the result properly, we need further terminology. Denoting the collection of structure coefficients ηi1 ,...,in ,k by y ∈ CN , we write p(x) = Q(y, x). (12) Theorem 5. The set of all y such that Cn admits a basis of idempotents of p = Q(y, ·) contains a Zariski–open (and dense) subset of parameter space CN . Proof. We set F (y, x) := Q(y, x) − x. Then, for a fixed parameter y0 , an idempotent of Q(y0 , ·) is a nonzero solution of the equation Q(y0 , x) − x = 0, equivalently F (y0 , x) = 0. A. We first show the existence of a norm–open set in parameter space that satisfies the desired property. 1. We use the Implicit Function Theorem: Assume that x0 is a nonzero solution of F (y0 , x) = 0, and that the partial derivative Dx F (y0 , x0 ) = Dx Q(y0 , x0 ) − I is invertible. Then there exists an analytic function h from some neighborhood U of y0 to Cn such that h(y0 ) = x0 and Q(y, h(y)) − h(y) = 0, for all y ∈ U. Thus, for any small change of the structure coefficients there is a nearby idempotent. (The condition on the derivative, incidentally, means that the idempotent x0 is of multiplicity one.) 2. The same argument can be applied to a basis of idempotents: Assume there is a parameter (1) (n) y0 such that the equation F (y0 , x) = 0 admits n linearly independent solutions x0 , . . . , x0 , (i) and that the partial derivative with respect to x at (y, x0 ) is invertible. Then consider the map G : CN × (Cn )n → (Cn )n , (y, x(1) , . . . , x(n) ) 7→ (F (y, x(1) ), . . . , F (y, x(n) )). 18 (1) (n) Here we have a solution (y0 , x0 , . . . , x0 ) of G = 0 and the matrix representing the par- tial derivative with respect to (x(1) , . . . , x(n) ) is block diagonal with invertible blocks, hence invertible. Therefore there exist a neighborhood U of y0 and analytic maps (i) h(i) : U → Cn with h(i) (y0 ) = x0 and Q(y, h(i) (y)) − h(i) (y) = 0 for y ∈ U , 1 ≤ i ≤ n. Moreover   det h(1) (y), . . . , h(n) (y) does not vanish at y0 , hence in some open neighborhood U e ⊆ U of y0 . In other words, for e all y ∈ U the corresponding p = Q(y, ·) admits a basis of idempotents. 3. To finish the argument, one needs to exhibit one p (thus some y0 ) for which a basis of idempotents exists and the respective partial derivatives are invertible: Take  m x1  ..  ∗ p (x) =  .  =: Q(y ∗ , x), xmn and the standard basis e1 , . . . , en . B. Now we show the existence of a Zariski–open set in parameter space which has the desired properties. For 1 ≤ k < ` ≤ n define ∆k,` (u, v) := uk v` − u` vk ; u, v ∈ Cn . Consider the morphism n H : CN × (Cn )n → Cn(n−1)/2  × C,   ∆k,` (Q(y, x(i) ), x(i) )i,k,` y, x(1) , . . . , x(n) 7→ . det(x(1) , . . . , x(n) ) − 1 Then H = 0 defines an algebraic subvariety Z of CN × (Cn )n , which is nonempty by the first part of the proof. Any zero of H corresponds to some homogeneous p which admits a basis v1 , . . . , vn of Cn with p(vi ) ∈ Cvi . The image Z e of Z under the projection π : CN × (Cn )n → CN onto the first component contains a Zariski–open subset of its closure, by a standard theorem on morphisms of algebraic varieties (see e.g. Shafarevich [20], Chapter 1, Theorem 1.14). But e also contains a norm–open set U , whose Zariski closure is due to the first part of the proof, Z all of C . Therefore Z N e contains a nonempty Zariski–open subset of CN . Finally, those p possessing a nilpotent correspond to a proper Zariski–closed subset of pa- rameter space, thanks to a resultant argument; see e.g. Röhrl [17], Theorem 1. Taking the complement of this closed subset, we find that all p(vi ) ∈ Cvi \ {0}, whence some multiple of vi is an idempotent, and the proof is finished.  5.3. Construction of quadratic vector fields in dimension three: Some details. Here we describe some details of the construction of distinguished quadratic vector fields, and outline some arguments. We also supply, in the additional material, some Maple worksheets containing the calculations. 19 5.3.1. The explicit expression for system (9). The quadratic homogeneous vector field (9) having the prescribed six idempotents e1 , e2 , e3 , v1 , v2 , v3 with vi = γi1 e1 + γi2 e2 + γi3 e3 and i = 1, 2, 3 is given by  p1 = x1 2 + x1 x2 −γ11 2 γ21 γ23 γ32 γ33 + γ11 2 γ22 γ23 γ31 γ33 + γ11 γ13 γ21 2 γ32 γ33 − γ11 γ13 γ22 γ23 γ31 2 −γ12 γ13 γ21 2 γ31 γ33 + γ12 γ13 γ21 γ23 γ31 2 − γ11 γ13 γ21 γ32 γ33 + γ11 γ13 γ22 γ23 γ31 +γ11 γ21 γ23 γ32 γ33 − γ11 γ22 γ23 γ31 γ33 − γ12 γ13 γ21 γ23 γ31 + γ12 γ13 γ21 γ31 γ33 ) − x2 x3 γ11 γ21 γ31 (γ11 γ22 γ33 − γ11 γ23 γ32 − γ12 γ21 γ33 + γ12 γ23 γ31 + γ13 γ21 γ32 − γ13 γ22 γ31 −γ12 γ23 + γ12 γ33 + γ13 γ22 − γ13 γ32 − γ22 γ33 + γ23 γ32 ) + x1 x3 γ11 2 γ21 γ22 γ32 γ33 − γ11 2 γ22 γ23 γ31 γ32 − γ11 γ12 γ21 2 γ32 γ33 + γ11 γ12 γ22 γ23 γ31 2 +γ12 γ13 γ21 2 γ31 γ32 − γ12 γ13 γ21 γ22 γ31 2 + γ11 γ12 γ21 γ32 γ33 − γ11 γ12 γ22 γ23 γ31 −γ11 γ21 γ22 γ32 γ33 + γ11 γ22 γ23 γ31 γ32 + γ12 γ13 γ21 γ22 γ31 − γ12 γ13 γ21 γ31 γ32 )] /d  p2 = x2 2 + x1 x2 γ11 γ13 γ22 2 γ32 γ33 − γ11 γ13 γ22 γ23 γ32 2 − γ12 2 γ21 γ23 γ32 γ33 +γ12 2 γ22 γ23 γ31 γ33 + γ12 γ13 γ21 γ23 γ32 2 − γ12 γ13 γ22 2 γ31 γ33 + γ11 γ13 γ22 γ23 γ32 −γ11 γ13 γ22 γ32 γ33 − γ12 γ13 γ21 γ23 γ32 + γ12 γ13 γ22 γ31 γ33 + γ12 γ21 γ23 γ32 γ33 −γ12 γ22 γ23 γ31 γ33 ) + x2 x3 −γ11 γ12 γ21 γ23 γ32 2 + γ11 γ12 γ22 2 γ31 γ33 + γ11 γ13 γ21 γ22 γ32 2 − γ11 γ13 γ22 2 γ31 γ32 −γ12 2 γ21 γ22 γ31 γ33 + γ12 2 γ21 γ23 γ31 γ32 + γ11 γ12 γ21 γ23 γ32 − γ11 γ12 γ22 γ31 γ33 −γ11 γ13 γ21 γ22 γ32 + γ11 γ13 γ22 γ31 γ32 + γ12 γ21 γ22 γ31 γ33 − γ12 γ21 γ23 γ31 γ32 ) − x1 x3 γ12 γ22 γ32 (γ11 γ22 γ33 − γ11 γ23 γ32 − γ12 γ21 γ33 + γ12 γ23 γ31 +γ13 γ21 γ32 − γ13 γ22 γ31 + γ11 γ23 − γ11 γ33 − γ13 γ21 + γ13 γ31 + γ33 γ21 − γ23 γ31 )] /d p3 = x3 2 − [x1 x2 γ13 γ23 γ33 (γ11 γ22 γ33 − γ11 γ23 γ32 − γ12 γ21 γ33 + γ12 γ23 γ31 + γ13 γ21 γ32 − γ13 γ22 γ31 − γ11 γ22 + γ32 γ11 + γ12 γ21 − γ31 γ12 − γ21 γ32 + γ31 γ22 ) +x1 x3 γ11 γ12 γ22 γ23 γ33 2 − γ11 γ12 γ23 2 γ32 γ33 − γ12 γ13 γ21 γ22 γ33 2 + γ12 γ13 γ23 2 γ31 γ32 +γ13 2 γ21 γ22 γ32 γ33 − γ13 2 γ22 γ23 γ31 γ32 − γ11 γ12 γ22 γ23 γ33 + γ11 γ12 γ23 γ32 γ33 +γ12 γ13 γ21 γ22 γ33 − γ12 γ13 γ23 γ31 γ32 − γ13 γ21 γ22 γ32 γ33 + γ13 γ22 γ23 γ31 γ32 ) −x2 x3 γ11 γ12 γ21 γ23 γ33 2 − γ11 γ12 γ23 2 γ31 γ33 − γ11 γ13 γ21 γ22 γ33 2 + γ11 γ13 γ23 2 γ31 γ32 +γ13 2 γ21 γ22 γ31 γ33 − γ13 2 γ21 γ23 γ31 γ32 − γ11 γ12 γ21 γ23 γ33 + γ11 γ12 γ23 γ31 γ33 +γ11 γ13 γ21 γ22 γ33 − γ11 γ13 γ23 γ31 γ32 − γ13 γ21 γ22 γ31 γ33 + γ13 γ21 γ23 γ31 γ32 )] /d with d = det(A) = γ11 γ12 γ21 γ23 γ32 γ33 − γ11 γ12 γ22 γ23 γ31 γ33 − γ11 γ13 γ21 γ22 γ32 γ33 +γ11 γ13 γ22 γ23 γ31 γ32 + γ12 γ13 γ21 γ22 γ31 γ33 − γ12 γ13 γ21 γ23 γ31 γ32 . See the Maple worksheet ExplicitSystem. 5.3.2. Concerning the proof of Proposition 4. For the values γ11 = −1, γ12 = 3, γ13 = 2, γ21 = 1, γ22 = 1, γ23 = −2, γ31 = 0, γ32 = 1, γ33 = −3 the Jacobian of Γ is invertible; see Maple worksheet JacobianInvertible. 5.3.3. Computing the seventh idempotent. To calculate the coefficients of the seventh idempotent v = s1 e1 + s2 e2 + s3 e3 in terms of the nine parameters γij , first rewrite the components of the vector field into the form p1 = x21 + A1 (x2 , x3 )x1 + A2 (x2 , x3 ), p2 = B1 (x2 , x3 )x1 + B2 (x2 , x3 ), p3 = C1 (x2 , x3 )x1 + C2 (x2 , x3 ), 20 with A1 = a11 x2 + a12 x3 − 1, A2 = a13 x2 x3 , B1 = b11 x2 + b12 x3 , B2 = x22 − x2 + b13 x2 x3 , C1 = c11 x2 + c12 x3 , C2 = x23 − x3 + c13 x2 x3 . (This is a preliminary step to avoid huge expressions involving the γij in Maple worksheets.) The idempotents are the nonzero solutions of the equation p(x) − x = 0 where x = (x1 , x2 , x3 ). So we obtain x21 + A1 x1 + A2 = 0, B1 x1 + B2 = 0, C1 x1 + C2 = 0. Here we may substitute x1 = −B2 /B1 (or x1 = −C2 /C1 .) From the former one obtains two equations B1 C2 − C1 B2 = 0, B22 − A1 B1 B2 + A2 B12 = 0 in variables x2 , x3 . Compute the resultant of this system with respect to the variable x2 . This is a polynomial of degree 12 in the variable x3 , of the form R(x3 ) = x53 (x3 − 1)b212 (b11 b13 x3 − b12 x3 − b11 )2 Te4 (x3 ) with Te4 a polynomial of degree 4 in the variable x3 . By construction, the third entries of the idempotents are among the zeros of R(x3 ). Here e1 , e2 and e3 correspond to a simple zero 1 and a double zero 0, and one verifies that the linear factor b11 b13 x3 − b12 x3 − b11 does not correspond to the third entry of any vi . So, the third entries of v1 , v2 and v3 are roots of Te. One may compute the fourth root of this degree four polynomial as follows: First normalize (divide by the leading coefficient) to obtain the monic polynomial T4 . For T4 , the sum of the four roots is the negative coefficient of x33 . Since we know the three roots γ13 , γ23 , γ33 of T4 we are able to calculate the fourth one, which is rational in the γij , after re-substitution. We take this as a candidate for s3 . Similarly we may consider the resultant with respect to the variable x3 , which is a polynomial of degree 12 in the variable x2 and takes the form R( x2 ) = x52 (x2 − 1)b12 (b11 b13 x2 − b12 x2 + b12 )2 Se4 (x2 ), and proceed as above, finding a candidate for s2 . Finally use x1 = −B2 /B1 to obtain a candidate for s1 , and verify that this indeed yields an idempotent by direct calculation; see Maple worksheet 7thidempotent. It should be noted that the output is generally quite voluminous. 5.3.4. On the proofs of Propositions 5 and 6. It is sufficient to prove the latter, which is done by in- spection of the eigenvalues of the Jacobians of all idempotents; see Maple worksheet TestPropertyE. 5.3.5. Details for Example 1. Here we provide some details for the homogeneous quadratic vector field given in (10), with parameters √ √ √ √ √ √ γ11 = 2, γ12 = 3, γ13 = 0, γ21 = 0, γ22 = 3, γ23 = 5, γ31 = 2, γ32 = 0, γ33 = 5. In this case the output of the computations is of moderate size, so we can reproduce it here; see Maple worksheet Algebraicallydependent for the calculations.√ √ √ √ √ √ The system has the seven idempotents e1 , e2 , e3 , v1 = ( 2, 3, 0), v2 = (0, 3, 5), v3 = ( 2, 0, 5) and v = (s1 , s2 , s3 ), with A1 A2 s1 = − , A3 A4 √ √ √ √ √ √ √ −243 5 + 480 3 + 369 5 − 900 2 + 282 5 − 720 3 − 504 5 + 1020 s2 = √ √ √ √ √ √ √ , 440 5 √ − 1000 3√− 801 5√+ 1785 2√+ −673 √5 + 1495 √ 3 + 1191 √ 5 − 2685 −345 5 + 695 3 + 495 5 − 1245 2 + 380 5 − 950 3 − 780 5 + 1530 s3 = √ √ √ √ √ √ √ , 440 5 − 1000 3 − 801 5 + 1785 2 + −673 5 + 1495 3 + 1191 5 − 2685 21 and the abbreviations √ √ √ √ √ √ √ √ √ √ √ √  A1 = √ 5 − 573 362√ 2√ 3 √ √2 5 − 545 √ √ 3 5 − 780 √ √ 2 3 + 969 √ 5 + 1335 √ 1265 3 − 2085 , 2 +√ A2 = 69 √2 √3 √5 − 99 2√ 5√− 76 3 √5 √ − 139 2 √3 √ + 156 5 √+ 249 2 √+ 190 3 √− 306 ,  A3 = 133 √2√3√5 − 231 √2√5 − 208 √3√5 − 285 √2 √ 3 + 348 5√+ 543 2 √ + 484 3 −√744 ,  A4 = 440 2 3 5 − 801 2 5 − 673 3 5 − 1000 2 3 + 1191 5 + 1785 2 + 1495 3 − 2685 . The eigenvalues of the Jacobian matrix at the idempotents e1 , e2 and e3 are respectively √ √ √ √   − √5 − 1 √2/2, − √2 √3 − 1 /2, 2 , − √ 5√− 1 3/3, − √3 √2 − 1 /3, 2  , − 5 3 − 1 /5, − 5 2 − 1 /5, 2 . The eigenvalues of the Jacobian matrix at the idempotents v1 , v2 and v3 are respectively √ √ √  √ √ √  √ √ √  2, 2 + 3, −2 5 + 2 , 2, −2 2 + 2, 5 + 3 , 2, −2 3 + 2, 5 + 2 . The Jacobian matrix at the last idempotent v = (s1 , s2 , s3 ) has 2 as an eigenvalue and the other two are √ √ √ √ √ √ 794929 2 + 762999 3 + 880620 2 + 1744545 5 1796931 2 + 3382333 3 λ± = + √ √ 4061514 4061514 A 776393 2 3211015 ± + + , 4061514 676919 1353838 with √ √ √ √ A = −6061791842292√ 5 + 20403579754296 √ 3 − 10627847112816 √ 5 + 45521293739166 2 + −8804787537402 5 + 29950278886104 3 − 15500011528278 5 + 66711726928548. In each case, inspection shows that the eigenvalues are linearly independent over the rational number field Q. Acknowledgements. NK acknowledges support by the DFG Research Training Group GRK 1632 “Experimental and Constructive Algebra”. JL is partially supported by the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. JL and CP are also supported by the Ministerio de Ciencia, Inno- vación y Universidades, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER). CP is additionally partially supported by the Catalan Grant 2017SGR1049 and the Spanish MINECO- FEDER Grants MTM2015-65715-P and PGC2018-098676-B-I00/AEI/FEDER/UE. Finally, CP and SW thank the CRM at Universitat Autónoma de Barcelona for its hospitality during a re- search visit in May and June 2019. References [1] Berrone, Lucio R.; Giacomini, Hector: Inverse Jacobi multipliers. Rend. Circ. Mat. Palermo (2) 52 (2003), no. 1, 77–130. [2] Bibikov, Yuri N.: Local theory of nonlinear analytic ordinary differential equations. 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Doctor Marañón, 44–50, 08028 Barcelona, Catalonia, Spain E-mail address: chara.pantazi@upc.edu 23