DEGREE OF THE FIRST INTEGRAL OF A PENCIL IN P 2 DEFINED BY LINS NETO

Let P4 be the linear family of foliations of degree 4 in P2 introduced by A. Lins Neto, whose set of parameter with first integral Ip(P4) is dense and countable. In this work, we will compute explicitly the degree of the rational first integral of the foliations in this linear family, as a function of the parameter. 2010 Mathematics Subject Classification: 34A26, 32S65, 37F75.


Introduction
One of the main problems in the theory of planar vector fields is to characterize the ones which admit a first integral.The invariant algebraic curves are a central object in integrability theory since 1878, year when Darboux found connections between algebraic curves and the existence of first integrals of polynomial vector fields.Thus, the first question was to know if a polynomial vector field has or not invariant algebraic curves, which was partially answered by Darboux in [4].The most important improvements of Darboux's results were given by Poincaré in 1891, who tried to answer the following question: "Is it possible to decide if a foliation in P 2 has a rational first integral?"This problem is known as the Poincaré Problem.In [14], he observed that it is sufficient to bound the degree of a possible algebraic solution.By imposing conditions on the singularities of the foliation he obtains necessary conditions which guarantee the existence of a rational first integral.More recently, this problem has been reformulated as follows: given a foliation on P 2 , try to bound the degree of the generic solution using information depending only on the foliation, for example on its degree or on the eigenvalues of its singularities.
The work of this author was partially supported by CNPQ.
In 2002, Lins Neto [9] built some notable linear families of foliations (which he later called pencil [10]) in P 2 , where the set of parameters in which the foliation has a first integral is dense and countable.The importance of these families is that there is no bound depending only on the degree and the analytic type of their singularities.One of such families is the pencil It is well known that I p (P 4 ), the set of parameters of foliations in P 4 which have a first integral, is the imaginary quadratic field Q(τ 0 ), where The purpose of this work is to compute the degree of the foliations F α in P 4 with rational first integral as a function of α.For this, we first relate the pencil P 4 with a pencil of linear foliations P * 4 in a complex torus E × E, where E = C/ 1, τ 0 .Then we derive the formula of the degree using the ideal norm of the ring Z[τ 0 ] as sketched below.Consequently, we are able to address the Poincaré Problem for the foliations in P 4 .
Given a foliation F t ∈ P 4 , with t ∈ I p (P 4 ) there exists a unique corresponding foliation G α(t) ∈ P * 4 where α(t) = t−1 −2−τ0 .Then writing α(t) = β1 α1 , with α 1 , β 1 ∈ Z[τ 0 ] and (α 1 , β 1 ) = 1, we have proved the following result: Furthermore, we show that the growth of the counting function π P4 (see Section 4.2), which associates to every n ∈ N, the number of elements in I p (P 4 ) for which the corresponding foliation has a first integral of degree at most n, has quadratic order, that is This is an improvement over a previous result due to Pereira [13,Proposition 4].

Preliminaries
From now on, τ 0 will denote the complex number e 2πi/3 .Let F be a foliation associated with the 1-form ω.Given a singularity p of F, let λ 1 and λ 2 be the eigenvalues of the linear part of the vector field associated to ω. Recall that p is of type (a : b) if [λ 1 : λ 2 ] = [a : b] ∈ P 1 .Moreover, if p is of type (1 : 1) then p is called a radial singularity.Let us state some properties of the pencil P 4 : (1) The tangency set of P 4 , given by ω ∧ η = 0, is the algebraic curve This curve is composed by nine invariant lines such that the set of intersections of these lines is formed by twelve points.We will denote such lines and points by L = {L 1 , . . ., L 9 } and P = {e 1 , . . ., e 12 }, respectively.
Let P * 4 , using φ, the pencil X 0 induced by the linear pencil in C 2 (1) Let ϕ : X 0 → X 0 be the holomorphic map defined by ϕ(x, y) = (τ 0 x, τ 0 y).Denote as Fix(ϕ), the set fixed points of ϕ.Consider p 1 = 0, (2) Fix(ϕ) has nine elements, namely, (p l , p k ) 3 l,k=1 .Now consider the four elliptic curves in X 0 , and let C be the set of these curves.For F ∈ C and p ∈ Fix(ϕ), denote F p = F + p.Hence, the set E := {F p : p ∈ Fix(ϕ), F ∈ C } consists of twelve elliptic curves.Since ϕ(F p ) = F p and Fix(ϕ) ∩ F p = (Fix(ϕ) ∩ F ) + p, we conclude that two different elliptic curves intersect only in three fixed points of ϕ.
3. Relation between the pencils P * 4 and P 4 The relation between the pencils P * 4 and P 4 was given by McQuillan in [12], where he proved the following result.
We now give an idea of how the function g is constructed.We refer the reader to [15] for the details.Let π : X → X 0 be obtained from X 0 by blowing-up the nine fixed points of ϕ, and denote then Y is a smooth rational surface.Moreover, the quotient map h : X → Y is a finite morphism with degree 3 and its ramification divisor is R = ) is a biholomorphism, the rational map h maps D i onto a rational curve D i , with self-intersection −3, for i = 1, . . ., 9. Furthermore, the map h sends π * E i , the strict transformation of E i , into a rational curve denoted by E i , with self-intersection −1, as shown in Figure 3.
Let π 1 : Y → Y 0 be the blowing-down of the curves E 1 , . . ., E 12 .The following lemma holds.Lemma 2. With the previous notations we have that Y 0 = P 2 .
Let g the rational map defined by (see Figure 4) Then g contracts each elliptic curve of C * into a point in P 2 and sends each point of C * into an algebraic curve L in P 2 with self-intersection one.In particular, L is a line in P 2 .Thus, the configuration of lines and points g(C * ) consists in twelve points and nine lines of P 2 , denoted by E * and Fix(ϕ) * , respectively, satisfying the following properties: (1) each line in Fix(ϕ) * contains four points of E * ; (2) each point of E * belongs to two lines of Fix(ϕ) * ; (3) if three points of E * are not in a line in Fix(ϕ) * then the points are not aligned.
We will see that is easier to obtain properties of the leaves of P * 4 (see Section 3.1), because they are elliptic curves.
Proposition 5. Let P = {H α } α∈C be a pencil of linear foliations in X, as above.Then In particular, ).To calculate the degree of the first integral of a foliation in F 4 α it is sufficient to calculate the autointersection of two leaves of G α .Thus, we need to recall some properties of elliptic curves.
Let K ⊂ C be an algebraic number field and let O K be the ring of algebraic integers contained in K. Given an ideal I of O K , we consider the quotient ring O K /I which is finite (cf.[17, p. 106]).The ideal norm of I, denoted by N O K (I), is the cardinality of O K /I.
The Dedekind Zeta function of K is defined, for a complex number s with Re(s) > 1, by the Dirichlet series where I ranges through the non-zero ideals of the ring of integers O K of K.This sum converges absolutely for all complex numbers s with Re(s) > 1.Note that ζ Q coincides with the Riemann zeta function.
As an application let us consider the following example: 5), for α, β, γ, δ ∈ Z[τ 0 ] such that (α, β) = 1 and (γ, δ) = 1, the intersection number of the elliptic curves E α,β and E γ,δ is 4. Degree of the first integral of a foliation is the leaf of G α passing through (0, 0).Let F t be the rational first integral of F 4 t and d t be the degree of F t .We want to determine d t .For this, take a generic irreducible fiber C of F t , of degree d t .We can suppose that C * := g * (C) = E α1,β1 + p, where p / ∈ Fix(ϕ).Set C * 1,0 := E 1,0 + p in X 0 and let C 1,0 = g(C * 1,0 ) be the curve obtained in P 2 .The idea for computing d t is to find the relation between the intersection of C and C 1,0 in P 2 and the intersection of C * and C * 1,0 in X 0 (see Figure 5).
We observe that (7) Let C and C 1,0 the strict transforms of C and C 1,0 by π 1 , respectively, then where m p is the multiplicity of C in p and D p = π −1 1 (p).Furthermore, where E * ∩ C 1,0 = E * \ {e 1 , e 6 , e 8 }.
Remark 7. In (11), if we take α = a + τ 0 b and β = c + τ 0 d then In particular, d t is a multiple of 3.
4.2.The growth of the pencil P 4 .In [13], Pereira defines the counting function π C of an algebraic curve C included in Fol(2, d), the space of foliations in P 2 of degree d.Take P = {F α } α∈C a line in Fol(2, d), that is a pencil of foliations in P 2 .For n ∈ N, denote Thus, the counting function of P, Also in [13], the author observes the importance of study the function π P and shows the following example (cf.[13,Example 3]).
Example 8. Let P = {F α } α∈C be a pencil in P 2 , where F α is given by αx dy − y dx.

π 2 + 2 .
O n ln(n) 2/3 ln ln(n) Now, we will estimate π P4 (n), for n ∈ N, and see that the counting function π P 4 has the same behavior as in Example 8.