Characterization of second type plane foliations using Newton polygons
Percy Fernández-Sanchez, Evelia R. García Barroso, Nancy Saravia-Molina
CCHARACTERIZATION OF SECOND TYPE PLANE FOLIATIONSUSING NEWTON POLYGONS
PERCY FERN ´ANDEZ-S ´ANCHEZ, EVELIA R. GARC´IA BARROSO,AND NANCY SARAVIA-MOLINA
Abstract.
In this article we characterize the foliations that have the sameNewton polygon that their union of formal separatrices, they are the folia-tions called of the second type. In the case of cuspidal foliations studied byLoray [ Lo ], we precise this characterization using the Poincar´e-Hopf index.This index also characterizes the cuspidal foliations having the same desin-gularization that the union of its separatrices. Finally we give necessary andsufficient conditions when these cuspidal foliations are generalized curves, anda characterization when they have only one separatrix. Introduction
Camacho, Lins-Neto and Sad [
Cam-Li-Sad ] introduced and studied the singular-ities of foliations of the generalized curve type, these are the foliations withoutsaddle-nodes in their reduction of singularities. These foliations receive this namebecause they have a behavior similar to the union of their separatrices, where aseparatrix is an irreducible analytical curve invariant for the foliation. For thesefoliations the Poincar´e-Hopf index coincides with the Milnor number of the unionof their separatrices ([
Mol-So ] and [
Cam-Li-Sad ]) and the reduction of singular-ities of these foliations coincides with the desingularization of the union of theirseparatrices.The singularities of generalized curved type verify that their G´omez Mont - Seade -Verjovski index [
Go-Sea-Ve ] is zero and their Camacho - Sad index [
Cam-Sad2 ]and the Baum-Bott index are equal [ Br ].The foliations of the second type can be thought of as a generalization of the singu-larities of the generalized curve type, in which we will allow the existence of formalseparatrices. In order to add the formal separatrix we must admit that we havesaddle-nodes in their resolution of singularities that generate formal separatrices,they could not be a corner of two divisors, nor could saddle-nodes outside the cor-ners with weak separatrix contained in the divisor. Note that with these restrictionsthe singularities of a second type foliation with a single separatrix will have to bea generalized curve foliation. The singularities of second type were introduced byMattei and Salem [ Ma-Sal ]. They characterized this type of singularities by meansof the coincidence of the multiplicity of the foliation with the multiplicity of the
Date : December 18, 2018.First-named and third-named authors were partially supported by the Pontificia UniversidadCat´olica del Per´u project VRI-DGI 2017-01-0083. Second-named author was partially supportedby the Spanish grant MTM2016-80659-P. a r X i v : . [ m a t h . D S ] D ec PERCY FERN´ANDEZ-S´ANCHEZ, EVELIA R. GARC´IA BARROSO, AND NANCY SARAVIA-MOLINA union of their formal separatrices. For these singularities, the reduction of singular-ities coincides with the desingularization of its separatrix. It should be noted thatthe proof made in [
Cam-Li-Sad ] to prove this property for generalized curve typefoliations also proves this property for second type singularities. There are othercharacterizations of these singularities (see [
Can-Co-Mol ] and [
FP-Mol ]).Merle [
Mer ] gives a decomposition of the polar curve of an irreducible curve C that determines the topology of C . This theorem was generalized for foliationsby Rouill´e [ R ], where he gives a decomposition of the polar of a foliation, of thegeneralized curve type, which determines the topology of its separatrix. The mainingredient for his decomposition is Dulac’s Theorem [ Du ], this theorem tells usthat the Newton polygon of the foliation coincides with the Newton polygon of theseparatrix. Merle’s theorem has been generalized for reduced curves by [ GB ] and[ GB-Gw ], the first of which was generalized for foliations by [ Co ] and the secondby [ Sar ]. There are examples of foliations where their Newton polygon coincideswith that of their separatrices and is not a foliation of the generalized curve type,however these foliations are of the second type (see Example 3.3). In this paper wegive a new characterization of the singularities of the second type in terms of theNewton polygon of their union of separatrices.
Theorem 1.1.
A non-dicritical foliation is of the second type if and only if itsNewton polygon coincides with that of their union of separatrices.
We will prove Theorem 1.1 in Section 4.According to Loray [ Lo ], a foliation with a cuspidal singularity is given by F ω p,q, ∆ : ω p,q, ∆ = d( y p − x q ) + ∆( x, y )( px d y − qy d x ) , (1)where p, q are positive natural numbers and ∆( x, y ) ∈ C { x, y } .We will use Theorem 1.1 to characterize, in terms of the Poincar´e-Hopf index forfoliations, when the foliations with cuspidal singularities studied by Loray [ Lo ] areof the second type.Denote by PH( F ) the Poincar´e-Hopf index of the foliation F . For the foliationd( y p − x q ) we have PH ( p,q ) := PH(d( y p − x q )) = ( p − q − Theorem 1.2.
Let F ω p,q, ∆ be a cuspidal foliation as in (1) and suppose that F ω p,q, ∆ is non-dicritical. The next statements are equivalents: ( a ) The cuspidal foliation F ω p,q, ∆ is of the second type. ( b ) The intersection number (∆ , y p − x q ) ≥ PH ( p,q ) − . ( c ) The cuspidal foliation F ω p,q, ∆ has the same reduction of singularities that d( y p − x q ) . In general, if a foliation has the same reduction of singularities as its union ofseparatrices then the foliation is not of the second type (see Example 2.3). However,after Theorem 1.2, for the cuspidal foliations this property characterizes foliationsof the second type.We also give, in the next theorem, necessary and sufficient conditions when thecuspidal foliations are of the generalized curve type.
Theorem 1.3.
Let F ω p,q, ∆ be a cuspidal foliation as in (1) and suppose that F ω p,q, ∆ is non-dicritical. We have: HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS3 (a) If the intersection number (∆ , y p − x q ) > PH ( p,q ) − , then F p,q, ∆ ( x, y ) isof the generalized curve type.(b) If p, q are coprime then the foliation F ω p,q, ∆ is of generalized curve type ifand only if (∆ , y p − x q ) > PH ( p,q ) − . We will prove Theorem 1.2 and Theorem 1.3 in Section 5.2.
Basic Definitions and Notations
In order to fix the notation, we will remember basic concepts of local foliationtheory and plane curves. Denote by C [[ x, y ]] the ring of formal powers series intwo variables with coefficients in C and C { x, y } the sub-ring of C [[ x, y ]] formedby formal powers series that converge in a neighborhood of 0 ∈ C . Consider themaximal ideals m and (cid:98) m of C { x, y } and C [[ x, y ]] respectively. The order of a powerseries (cid:98) h ( x, y ) = (cid:88) ij a ij x i y j ∈ C [[ x, y ]] is ord( (cid:98) h ) := min { i + j : a ij (cid:54) = 0 } .A singular formal foliation (cid:99) F ω of codimension one over C is locally given by a1-form (cid:98) ω = (cid:98) A ( x, y )d x + (cid:98) B ( x, y )d y , where (cid:98) A, (cid:98) B ∈ (cid:98) m are coprime. The power series (cid:98) A and (cid:98) B are called the coefficients of (cid:98) ω . The multiplicity of the foliation (cid:99) F ω isdefined as mult( (cid:98) ω ) := min { ord( (cid:98) A ) , ord( (cid:98) B ) } .Let T ⊆ N . Denote by D ( T ) the convex hull of ( T + R ≥ ), where + is theMinkowski sum, and by N ( T ) the polygonal boundary of D ( T ), which will call Newton polygon determined by T .Let (cid:98) h ( x, y ) = (cid:88) i,j a ij x i y j ∈ C [[ x, y ]]. The support of (cid:98) h issupp( (cid:98) h ) := { ( i, j ) ∈ N : a ij (cid:54) = 0 } , and the Newton polygon of (cid:98) h is by definition the Newton polygon N (supp( (cid:98) h )).Let (cid:98) ω = (cid:98) A ( x, y )d x + (cid:98) B ( x, y )d y be a one-form, where (cid:98) A, (cid:98) B ∈ (cid:98) m . The support of (cid:98) ω issupp( (cid:98) ω ) = supp( x (cid:98) A ) ∪ supp( y (cid:98) B ) . If we write (cid:98) ω = (cid:88) i,j (cid:98) ω ij , where (cid:98) ω ij = (cid:98) A ij x i − y j d x + (cid:98) B ij x i y j − d y , thensupp( (cid:98) ω ) = { ( i, j ) : ( (cid:98) A ij , (cid:98) B ij ) (cid:54) = (0 , } . Let (cid:99) F ω : (cid:98) ω = 0 be a foliation given by the one-form (cid:98) ω . The Newton polygon of (cid:99) F ω ,denoted by N ( (cid:99) F ω ) or N ( (cid:98) ω ) is the Newton polygon N (supp( (cid:98) ω )).Let (cid:98) f ( x, y ) ∈ C [[ x, y ]]. We say that the (cid:98) S f : (cid:98) f ( x, y ) = 0 is invariant by (cid:99) F ω if (cid:98) ω ∧ d (cid:98) f = (cid:98) f . (cid:98) η, where (cid:98) η is a two-form (i.e. (cid:98) η = (cid:98) g d x ∧ d y , for some (cid:98) g ∈ C [[ x, y ]]). If (cid:98) S f is irreducible then we will say that (cid:98) S f is a formal separatrix of (cid:99) F ω : (cid:98) ω = 0.We will consider non-dicritical foliations, that is, foliations having a finite set ofseparatrices (see [ Cam-Li-Sad , page 158 and page 165]). Let ( (cid:98) S f j ) rj =1 be the setof all formal separatrices of the non-dicritical foliation (cid:99) F ω : (cid:98) ω = 0. Each separatrix PERCY FERN´ANDEZ-S´ANCHEZ, EVELIA R. GARC´IA BARROSO, AND NANCY SARAVIA-MOLINA (cid:98) S f j corresponds to an irreducible formal power series (cid:98) f j ( x, y ). Denote by (cid:98) S ( (cid:99) F ω )the union (cid:83) (cid:98) S f j of all separatrices of the foliation (cid:99) F ω , which we will call union offormal separatrices of (cid:99) F ω . In the following we will denote by F ω a holomorphicfoliation and by S ( F ω ) its union of convergent separatrices.The dual vector field associated to (cid:99) F ω is X = (cid:98) B ( x, y ) ∂∂x − (cid:98) A ( x, y ) ∂∂y . We saythat the origin ( x, y ) = (0 ,
0) is a simple or reduced singularity of (cid:99) F ω if the matrixassociated with the linear part of the field ∂ (cid:98) B (0 , ∂x ∂ (cid:98) B (0 , ∂y − ∂ (cid:98) A (0 , ∂x − ∂ (cid:98) A (0 , ∂y (2)has two eigenvalues λ, µ , with λµ (cid:54)∈ Q + .It could happen that a ) λµ (cid:54) = 0 and λµ (cid:54)∈ Q + in which case we will say that the singularity is notdegenerate or b ) λµ = 0 and ( λ, µ ) (cid:54) = (0 ,
0) in which case we will say that the singularity isa saddle-node .In the b ) case, the strong separatrix of a foliation with singularity P is an analyticinvariant curve whose tangent at the singular point P is the eigenspace associatedwith the non-zero eigenvalue of the matrix given in (2). The zero eigenvalue isassociated with a formal separatrix called weak separatrix .From now on π : M → ( C ,
0) represents the process of singularity reduction ordesingularization of (cid:99) F ω [ Ma-Mou ], obtained by a finite sequence of point blows-up, where D := π − (0) = n (cid:91) j =1 D j is the exceptional divisor , which is a finite unionof projective lines with normal crossing (that is, they are locally described by oneor two regular and transversal curves). In this process, any separatrix of (cid:99) F ω issmooth, disjoint and transverse to D j ⊂ D , and it does not pass through a corner(intersection of two components of the divisor D ). Let (cid:98) F ω be a non-dicritical for-mal foliation and consider the minimal reduction of singularities π : M → ( C , (cid:98) F ω (this is, a reduction with the minimal number of blows-up that reduces thefoliation). The strict transform of the foliation (cid:98) F ω is given by (cid:98) F (cid:48) ω = π ∗ (cid:98) F ω and the exceptional divisor is D = π − (0).A foliation (cid:99) F ω is a generalized curve if in its reduction of singularities there are nosaddle-node points.If in the desingularization of (cid:98) F ω , the exceptional divisor D at point P contains theweak invariant curve of the saddle-node, then the singularity is called saddle-nodetangent . Otherwise we will say that (cid:98) F ω is a saddle-node transverse to D at point P . HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS5
Definition 2.1.
The foliation (cid:98) F ω is of the second type with respect to the divisor D if no singular points of (cid:98) F (cid:48) ω is of tangent node type.Non-dicritical foliations of the second type were studied by Mattei and Salem[ Ma-Sal ], also by Cano, Corral and Mol [
Can-Co-Mol ] and in the dicritical caseby Genzmer and Mol [
Ge-Mol ] and Fern´andez P´erez-Mol [
FP-Mol ]. Mattei andSalem gave the next characterization of foliations of the second type in terms ofthe multiplicity of their union of formal separatrices:
Theorem 2.2. [ Ma-Sal , Th´eor`eme 3.1.9]
Let (cid:99) F ω be a non-dicritical foliation andlet (cid:98) S ( (cid:99) F ω ) : (cid:98) f ( x, y ) = 0 be a reduced equation of its union of separatrices. Considerthe minimal reduction of singularities π : ( M, D ) → ( C , of (cid:99) F ω . Then(1) π is a reduction of singularities of (cid:98) S ( (cid:99) F ω ) . Furthermore, if (cid:99) F ω is of thesecond type then π is the minimal reduction of singularities of (cid:98) S ( (cid:99) F ω ) .(2) mult( (cid:98) ω ) ≥ mult( (cid:98) S ( (cid:99) F ω )) and the equality holds if and only if (cid:99) F ω is of thesecond type. The reciprocal of the first statement of Theorem 2.2 is not true, that is, if the re-duction of singularities of the foliation and that of its union of separatrices coincidedoes not guarantee that the foliation is of the second type, as shown in the followingexample.
Example . The union of the separatrices of the foliation F ω = ( xy + y )d x − x d y is S ( F ω ) = xy . The foliation F ω and its union of separatrices are desingularizedafter a blow-up but the foliation is not of the second type because the strongseparatrix that passes through the saddle-node is not contained in the divisor.3. Foliations and Newton polygones
Non-dicritical generalized curve foliations are those in which no saddle-node pointsappear in their desingularization [
Sei ] and they have a finite number of separatrices[
Cam-Li-Sad ]. These foliations were studied by Camacho, Lins Neto and Sad whoproved that
Theorem 3.1. [ Cam-Li-Sad , Theorem 2]
Let F ω be a non-dicritical generalizedcurve foliation and S ( F ω ) its union of separatrices. Then F ω and S ( F ω ) have thesame reduction of singularities. Rouill´e obtained the following result on non-dicritical generalized curve foliations.In [ R ] it is indicated that Mattei reported that this result was known by Dulac[ Du ]. Proposition 3.2. [ R , Proposition 3.8] Let F ω be a non-dicritical generalized curvefoliation and S ( F ω ) : f ( x, y ) = 0 be a reduced equation of its union of separatrices.Then N ( ω ) = N ( df ) . The reciprocal of Theorem 3.1 and Proposition 3.2 are not true, as the followingexample shows:
Example . Consider b (cid:54)∈ Q and the foliation defined by ω = (( b − xy − y )d x +( xy − bx + xy )d y . A reduced equation of its union of separatrices is f ( x, y ) = xy ( x − y ) = 0. The foliation F ω and the curve f ( x, y ) = 0 are desingularised PERCY FERN´ANDEZ-S´ANCHEZ, EVELIA R. GARC´IA BARROSO, AND NANCY SARAVIA-MOLINA after a blow-up. The Newton polygons N ( ω ) and N ( f ) are equal but F ω is not acurve generalized type foliation since a saddle-node point appears in its reductionof singularities.In [ Br , pag 532] was introduced the G´omez-Mont-Seade-Verjovsky index denoted by
GSV ( F ω , S ( F ω )), where F ω : ω = 0 and S ( F ω ) : f ( x, y ) = 0 is a reduced equationof union of convergent separatrices of F ω . For f ∈ C { x, y } , there are g, h ∈ C { x, y } ,with h and f coprime, and an analytic one-form η such that gω = h d f + f η . In[ Br ], Brunella defines GSV ( F ω , S ( F ω )) = 12 πi (cid:90) ∂ S ( F ω ) gh d (cid:18) hg (cid:19) , when S ( F ω ) : f = 0 is irreducible and ω = A ( x, y ) dx + B ( x, y ) dy . We get GSV ( F ω , S ( F ω )) = ord t (cid:18) Bf y ( γ ( t )) (cid:19) , where γ ( t ) is a parametrization of S ( F ω ). Now, we remember some results on theindex GSV ( F ω , S ( F ω )). Theorem 3.4. [ Cav-Le , Th´eor`eme 3.3][ Br , Proposition 7] Let F ω be a non-dicritical foliation and let S ( F ω ) : f ( x, y ) = 0 be a reduced equation of its union ofseparatrices. Then F ω is a generalized curve foliation if and only if GSV ( F ω , S ( F ω )) =0 . The next result on generalized curve foliations was obtained by Rouill´e [ R ] and itwill be very useful in this paper. Denote by C [[ t ]] the ring of formal power seriesin the variable t and coefficients in C , and C { t } the subring of C [[ t ]] of convergentpower series. Lemma 3.5. [ R , Lemme 3.7] Let F ω and F ω two non-dicritical generalized curvefoliations with the same union of separatrices. If γ ( t ) = ( x ( t ) , y ( t )) ∈ ( C { t } ) with γ (0) = 0 , then ord t γ ∗ ω = ord t γ ∗ ω . We deduce from Example 3.3 that there are foliations having the same polygonas their union of separatrices but they are not generalized curve foliations. Ourobjective in this paper is to characterize the foliations having the same Newtonpolygon that its union of separatrices. They will be the non-dicritical foliations ofthe second type.4.
Characterization of a foliation of the second type in terms ofthe Newton polygon
In this section, a new characterization of the second-type non-dicritical foliationsis given in terms of the Newton polygon of the foliation and that of its union ofseparatrices.
Lemma 4.1.
Let (cid:99) F ω be a non-dicritical foliation and (cid:98) f ( x, y ) = 0 be a reducedequation of its union of separatrices. If N ( (cid:98) ω ) = N ( (cid:98) f ) then (cid:99) F ω is a foliation of thesecond type. HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS7
Proof.
Consider the foliation (cid:99) F ω given by (cid:98) ω = (cid:88) i,j (cid:98) A ij x i − y j d x + (cid:88) i,j (cid:98) B ij x i y j − d y .Since mult( (cid:98) ω ) = min { ord( (cid:98) A ) , ord( (cid:98) B ) } thenmult( (cid:98) ω ) = min { i + j − i, j ) ∈ N ( (cid:98) ω ) } = min { i + j − i, j ) ∈ N (d (cid:98) f ) } = mult(d (cid:98) f ) . (3)From (3) and the second statement of Theorem 2.2 we finish the proof. (cid:3) As a consequence of Lemma 4.1 and Theorem 2.2 we conclude that if N ( (cid:98) ω ) = N ( (cid:98) f )then the foliation (cid:99) F ω and its union of separatrices (cid:98) S ( (cid:99) F ω ) have the same resolution.In the following proposition we generalize Lemma 3.5 to second type foliations. Proposition 4.2.
Let (cid:99) F ω be a non-dicritical second type foliation and (cid:98) S ( (cid:99) F ω ) itsunion of separatrices . If γ ( t ) = ( x ( t ) , y ( t )) ∈ ( C [[ t ]]) , with γ (0) = 0 , then ord t γ ∗ (cid:98) ω = ord t γ ∗ d (cid:98) f . Proof. If γ ( t ) = ( x ( t ) , y ( t )) is a parameterization of a separatrix of (cid:98) ω and d (cid:98) f then (cid:98) ω ( γ ( t )) .γ (cid:48) ( t ) = 0 = d (cid:98) f ( γ ( t )) .γ (cid:48) ( t ) and we conclude the proposition in such a case.Suppose now that γ ( t ) is not a parameterization of any separatrix of (cid:98) ω and d (cid:98) f .We proceed by induction on the number of blows-up n needed in the process of thedesingularization of the foliation (cid:99) F ω . First we suppose that the number of blows-upis n = 0. Then the foliations defined by the one-forms (cid:98) ω and d (cid:98) f are reduced. If (cid:99) F ω is a generalized curve foliation then the proposition follows from Lemma 3.5.Suppose now that (cid:99) F ω is a reduced foliation with a saddle-node. We can considerthe formal form of the saddle-node given by the equation: − y p +1 d x + (1 + λy p ) x d y with p ≥ λ ∈ C , which under a change of coordinates can be expressed as (see [ Cam-Sad2 , Page66]) (cid:98) ω = x (1 + λy p )d y − y p +1 d x, and the reduced equation of its union of formal separatrices is given by (cid:98) f ( x, y ) = xy .We can write γ ( t ) = ( x ( t ) , y ( t )) = ( t a n ( t ) , t b n ( t )), where a, b are positive integersand n i ( t ) are units of C [[ t ]] (that is n i (0) (cid:54) = 0 for i = 1 , γ ∗ (cid:98) ω = [ bt a + b − n ( t ) n ( t ) + t a + b n ( t ) n (cid:48) ( t )+ t a + b + pb − n ( t )( n ( t )) p +1 ( λb − a )+ λt a + b + pb n ( t )( n ( t )) p n (cid:48) ( t ) − t a + b + pb n (cid:48) ( t )( n ( t )) p +1 ]d t, so mult( γ ∗ (cid:98) ω ) = a + b −
1. On the other hand d (cid:98) f = y d x + x d y, hence γ ∗ d (cid:98) f = t b n ( t )( at a − n ( t ) + t a n (cid:48) ( t ))d t + t a n ( t )( bt b − n ( t ) + t b n (cid:48) ( t ))d t = [ t a + b − n ( t ) n ( t )( a + b ) + t a + b ( n (cid:48) ( t ) n ( t ) + n ( t ) n (cid:48) )]d t, so mult( γ ∗ d (cid:98) f ) = a + b −
1. Therefore, if (cid:99) F ω is a reduced foliation with a saddle-node, then ord t ( γ ∗ (cid:98) ω ) = ord t ( γ ∗ d (cid:98) f ). PERCY FERN´ANDEZ-S´ANCHEZ, EVELIA R. GARC´IA BARROSO, AND NANCY SARAVIA-MOLINA
Now we suppose that the foliations defined by the one-forms (cid:98) ω and d (cid:98) f are notreduced and n >
0. Let E be the blow-up at the origin ( x, y ) = (0 ,
0) given by E : ( x, t ) = ( x, xt ), so E ∗ (cid:98) ω = x m (cid:101)(cid:98) ω , where m is the multiplicity of (cid:98) ω and (cid:101)(cid:98) ω is thestrict transform of (cid:98) ω . Denote by (cid:101) γ the strict transformation of the curve γ by E .By induction hypothesis, we get,ord t (cid:101) γ ∗ (cid:101)(cid:98) ω = ord t (cid:101) γ ∗ (cid:102) d (cid:98) f . On the other hand we have γ ∗ (cid:98) ω = x ( t ) m (cid:101) γ ∗ (cid:101)(cid:98) ω, henceord t γ ∗ (cid:98) ω = mult( x ( t ))mult( (cid:98) ω ) + ord t (cid:101) γ ∗ (cid:101)(cid:98) ω. (4)Since the foliation (cid:99) F ω is of the second type, by Theorem 2.2, using the inductionhypothesis and replacing in the equation (4) we get ord t γ ∗ (cid:98) ω = ord t γ ∗ d (cid:98) f and wefinish the proof of the proposition. (cid:3) Proposition 4.2 was also given in [
Can-Co-Mol , Corollary 1], but with other proof.
Corollary 4.3.
Let (cid:99) F ω be a non-dicritical second type foliation and let (cid:99) S f be areduced equation of its union of formal separatrices. Then N ( (cid:98) ω ) = N (d (cid:98) f ) .Proof. Since (cid:99) F ω is a second type foliation, using Theorem 2.2, we have that (cid:99) F ω hasthe same reduction of singularities as its union of formal separatrices and mult( (cid:98) ω ) =mult(d (cid:98) f ). Reasoning analogously as in the proof given by Rouill´e [ R ] in order toprove the Proposition 3.2, and by Proposition 4.2, we finish the proof. (cid:3) As a consequence of the Corollary 4.3 we have:
Corollary 4.4.
Let (cid:100) F ω and (cid:100) F ω be two non-dicritical second type foliations. If (cid:100) F ω and (cid:100) F ω have the same union of formal separatrices, then N ( (cid:99) ω ) = N ( (cid:99) ω ) .Example . The foliation F ω given by ω = ( ny + x n )d x − x d y , n ≥ F ω is S ( F ω ) : x = 0.We observe that supp( ω ) = { (1 , , ( n + 1 , } and supp( f ) = { (1 , } , hence theNewton polygons of F ω and S ( F ω ) are different. xy N ( ω ) xy N ( f ) Example . Let us go back to Example 3.3. The second type foliation given by ω = (( b − xy − y ) dx + ( xy − bx + xy ) dy with − b, − b (cid:54)∈ Q + has as union ofseparatrices to S ( F ) = xy ( x − y ). We observe that polygons N ( ω ) and N ( f ) areequal. HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS9 xy (1,2)(2,1)(1,3)(2,2) N ( ω ) = N ( f ) Proof of Theorem 1.1.
It is an immediate consequence of Corollary 4.3 andLemma 4.1. (cid:3)
Theorem 1.1 gives a new characterization of the non-dicritical second type foliationsusing its Newton polygon. 5.
Cuspidal Foliations
Cuspidal foliations are inspired by nilpotent foliations.
A foliation F ω in ( C ,
0) iscalled a nilpotent singularity if it is generated by a vector field X with a non-zeronilpotent linear part (that is, the matrix associated with the linear part of the fieldis nilpotent). The nilpotent singularities were generalized to cuspidal singularitiesby Loray [ Lo ], as we shall see below.In this section we characterize when foliations with cuspidal singularities are of thesecond type in terms of weighted order. Furthermore, by means of the weightedorder, we give necessary and sufficient conditions for these foliations to be of gen-eralized curve type.Given p, q ∈ N ∗ , we define the weighted degree of a monomial x i y j asdeg ( p,q ) ( x i y j ) = ip + jq gcd( p, q ) , and the weighted order of a power series f ( x, y ) = (cid:88) i,j a i,j x i y j ∈ C { x, y } asord ( p,q ) ( f ( x, y )) = min (cid:110) deg ( p,q ) ( x i y j ) : a i,j (cid:54) = 0 (cid:111) . Remember that according to Loray [ Lo ], a foliation with a cuspidal singularity isgiven as in (1), that is by F ω p,q, ∆ : ω p,q, ∆ = d( y p − x q ) + ∆( x, y )( px d y − qy d x ) , where p, q are positive natural numbers and ∆( x, y ) ∈ C { x, y } .On the other hand, remember that PH ( p,q ) := PH(d( y p − x q )) = pq − p − q + 1.Cuspidal foliations are nilpotent foliations when p = 2.For Loray, the foliations F ω p,q, ∆ and d( y p − x q ) have the same resolution of sin-gularities if and only if ord ( p,q ) (∆) > pq − p − q gcd( p,q ) = PH ( p,q ) − p,q ) . Fern´andez, Mozo andNeciosup [ F-Moz-N ], find an imprecision in the characterization originally pro-posed by Loray. These authors mention that the condition is sufficient but notnecessary, as can be seen from the following example.
Example 1.
The foliation ω = d( y − x ) + axy (6 x d y − y d x ) with a (cid:54)∈ {− (6 r +1) ζ/r ∈ Q > y ζ = 1 } has the same resolution as the foliation d( y − x ) = 0 , butthe function ∆( x, y ) = axy satisfies ord (6 , ∆ = 3 , so the inequality ord (6 , ∆ > PH ( p,q ) − p,q ) does not hold. For d = gcd( p, q ), we have y p − x q = d (cid:89) i =1 ( y pd − ζ i x qd ) , and γ i ( t ) = ( t pd , A i t qd ) with A pd i = ζ i is a parameterization of S i : f i ( x, y ) =( y pd − ζ i x qd ), with ζ ∈ C , ζ d = 1. We get(∆ , y p − x q ) = (cid:88) i (∆ , f i ) = d · ord ( p,q ) (∆) , where ( f, g ) = dim C C { x, y } / ( f, g ) is the intersection number of f and g . Lemma 5.1.
If the cuspidal foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is a non-dicritical folia-tion, then S ( F ω p,q, ∆ ) = y p − x q = 0 is its union of separatrices.Proof. The curve S f : y p − x q = 0 is an invariant curve of the foliation F ω p,q, ∆ . Put α = ord(∆). Then mult( ω p,q, ∆ ) = min { q − , p − , α + 1 } . (5)Suppose that p < q . The multiplicity of the curve S f is p . If we assume that thecurve S f is not the only separatrix of the foliation F ω p,q, ∆ , then mult( S ( F ω p,q, ∆ )) >p . Using (5), we have mult( ω p,q, ∆ ) = min { p − , α + 1 } . We will study bothpossibilities:(i) If mult( ω p,q, ∆ ) = p − p − ω p,q, ∆ ) ≥ mult( S ( F ω p,q, ∆ )) − >p −
1, which is a contradiction.(ii) If mult( ω p,q, ∆ ) = α + 1 then α + 1 = mult( ω p,q, ∆ ) ≥ mult( S ( F ω p,q, ∆ )) − >p − , which is a contradiction since α + 1 ≤ p − F ω p,q, ∆ is S ( F ω p,q, ∆ ) = y p − x q .The same reasoning happens when p ≥ q and we conclude that S ( F ω p,q, ∆ ) = y p − x q . (cid:3) Proposition 5.2.
Suppose that the cuspidal foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is non-dicritical. If (∆ , y p − x q ) ≥ PH ( p,q ) − , with d = gcd( p, q ) , then the foliation F ω p,q, ∆ is of the second type.Proof. Suppose without lost of generality that p < q and ord∆ = i + j . Since(∆ , y p − x q ) ≥ PH ( p,q ) − i p + j q ≥ PH ( p,q ) −
1. After p < q we get i q + j q > i p + j q ≥ PH ( p,q ) − , so i + j > p − − pq > p − α = ord∆ ≥ p −
1. Since mult(d f ) = p − S ( F ω p,q, ∆ ) : f ( x, y ) = y p − x q = 0, using (5) we have mult( ω p,q, ∆ ) = p −
1. Hencemult( ω p,q, ∆ ) = mult(d f ) and we conclude that the foliation F ω p,q, ∆ is of the secondtype. (cid:3) Proposition 5.3.
Suppose that the cuspidal foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is non-dicritical. If F ω p,q, ∆ is the second type, then (∆ , y p − x q ) ≥ PH ( p,q ) − . HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS11
Proof.
From Lemma 5.1 we get S ( F ω p,q, ∆ ) = y p − x q . Put d := gcd( p, q ). The linecontaining the only compact side of Newton polygone N (d f ) is L : pd i + qd j = pqd .Since F ω p,q, ∆ is of the second type, using Theorem 1.1 we have N ( ω p,q, ∆ ) = N ( f ).Therefore, the line L also contains the only compact side of the Newton polygonof N ( ω p,q, ∆ ), that is any ( a, b ) ∈ supp( ω p,q, ∆ ) verifies a pd + b qd ≥ pqd . Suppose that∆( x, y ) = (cid:88) i,j a ij x i y j ∈ C { x, y } , then ω p,q, ∆ = − qx q − − q (cid:88) i,j a ij x i y j +1 d x + py p − + p (cid:88) i,j a ij x i +1 y j d y, and supp( ω p,q, ∆ ) = { ( q, , ( i + 1 , j + 1) } ∪ { (0 , p )( i + 1 , j + 1) } , for ( i, j ) ∈ supp(∆).If ( i + 1 , j + 1) ∈ supp( ω p,q, ∆ ) then ( i + 1) pd + ( j + 1) qd ≥ pqd , so we conclude that(∆ , y p − x q ) = ip + jq ≥ PH ( p,q ) − (cid:3) Proposition 5.4.
Suppose that the cuspidal foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is non-dicritical. The foliation F ω p,q, ∆ has the same reduction of singularities that d( y p − x q ) , if and only if, (∆ , y p − x q ) ≥ PH ( p,q ) − . Proof.
Suppose that (∆ , y p − x q ) ≥ PH ( p,q ) −
1. By Proposition 5.2 the foliation F ω p,q, ∆ is of the second type and by Theorem 2.2 we conclude that F ω p,q, ∆ andd( y p − x q ) have the same reduction of singularities.Suppose now that F ω p,q, ∆ and d( y p − x q ) have the same reduction of singularities.The curve y p − x q = 0 with p > q and d = gcd( p, q ) is desingularized by E : ( x, y ) = ( u n v pd , u m v qd ) , (6)such that mp − nq = d and m, n ∈ N ∗ . The transformation of ω p,q, ∆ = ( − qx q − − qy ∆( x, y ))d x + ( py p − + px ∆( x, y ))d y, by E defined as (6) is E ∗ ω p,q, ∆ = (cid:104) u nq − v pqd (cid:0) − nq + mpu mp − nq (cid:1) + dv pqd u nq − ( u m + n − nq v p + q − pqd E ∗ (∆( x, y ))) (cid:105) d u + (cid:104) pqd u nq v pqd − ( − u mp − nq ) (cid:105) d v = (cid:16) u nq − v pqd − (cid:17) v ( − qn + mpu d + (cid:101) ∆( u, v ))d u + pqd u ( u d − v, (7)where (cid:101) ∆( u, v )) = dE ∗ (∆( x, y )) u m + n − nq v p + q − pqd = (cid:88) i,j da ij u ni + mj + m + n − nq v pi + qj + p + q − pqd . Hence E ∗ ω p,q, ∆ u nq − v pq − = v ( − qn + mpu d + (cid:101) ∆( u, v ))d u + pqd u ( u d − v, (8)which singularities are (0 ,
0) and ( ζ j , ζ is a d th-primitive root of the unity.The dual vector field associated with the foliation defined by (8) is X = pqd ( u d − u ∂∂u − v ( − nq + mpu d + (cid:101) ∆( u, v )) ∂∂v , and the matrix associated with this field is DX = (cid:32) − pqd + ( d +1) pqd u d ∗ nq − mpu d − (cid:101) ∆( u, v ) − v ∂ (cid:101) ∆( u,v ) ∂v (cid:33) . (1) In (0 ,
0) we have DX = (cid:18) − pqd ∗ nq (cid:19) . Therefore, the singularity (0 ,
0) isnot degenerate.(2) If u d = 1 and v = 0 then we get DX = (cid:18) − pqd ∗ − d − (cid:101) ∆( u, v ) (cid:19) . Sincethe foliation is reduced, it could happen that • − d − (cid:101) ∆( ζ j ,
0) = 0, from where (cid:101) ∆( ζ j ,
0) = − d , in which case thesingularity is of saddle-node type. • − d − (cid:101) ∆( ζ j ,
0) = − a, so that λ = pq − a (cid:54)∈ Q + , in this case, the singularityis of a no degenerate type.We conclude that ord v (cid:101) ∆ ≥
0, so pi + qj + p + q ≥ i, j ). Therefore(∆ , y p − x q ) ≥ PH ( p,q ) − i, j ) ∈ supp(∆). (cid:3) Proof of Theorem 1.2.
The equivalence of statements ( a ) and ( b ) is a directconsequence of Propositions 5.2 and 5.3. The equivalence of statements ( b ) and ( c )is Proposition 5.4. (cid:3) Corollary 5.5.
Suppose that the cuspidal foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is non-dicritical. If the foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is of the generalized curve type then (∆ , y p − x q ) ≥ PH ( p,q ) − . The fact that the foliation F ω p,q, ∆ is of generalized curve type does not imply that(∆ , y p − x q ) > PH ( p,q ) −
1, as the next example shows:
Example 2.
The foliation ω = d ( y − x ) + axy (6 xdy − ydx ) , with a ∈ {− (6 r + 1) ζ/r ∈ Q > y ζ = 1 } ⊆ C ∗ , and a (cid:54) = − is of the generalizedcurve type, but (∆ , y p − x q ) = 3 = PH ( p,q ) − , where p = 6 , q = 3 and d = 3 . Nevertheless
Proposition 5.6.
Suppose that the cuspidal foliation F ω p,q, ∆ : ω p,q, ∆ = 0 is non-dicritical and p and q are coprimes. The foliation F ω p,q, ∆ is of generalized curvetype, if and only if (∆ , y p − x q ) > PH ( p,q ) − .Proof. Let us consider ω p,q, ∆ = ( − qx q − − qy ∆)d x + ( py p − + px ∆)d y , f ( x, y ) = y p − x q and γ ( t ) = ( t p , t q ) a parameterization of f ( x, y ) = 0. Thus GSV ( F ω p,q, ∆ , S ( F ω p,q, ∆ )) = ord t (cid:16) py p − + px ∆ py p − ( t p , t q ) (cid:17) = ord t (cid:16) t p ∆( t p ,t q ) t q ( p − (cid:17) , (9)where ∆( t p , t q ) = (cid:88) ij a ij t pi + qj . Note that GSV ( F ω p,q, ∆ , S ( F ω p,q, ∆ )) = 0 , if and only if ord t (cid:18) t p ∆( t p , t q ) t q ( p − (cid:19) = 0 , HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS13 what is equivalent to (∆ , y p − x q ) > PH ( p,q ) −
1. From Theorem 3.4 we concludethat F ω p,q, ∆ is of the generalized curve type, if and only if (∆ , y p − x q ) > PH ( p,q ) − (cid:3) Suppose now that p and q are not coprime. We will analyze if the strict inequality(∆ , y p − x q ) > PH ( p,q ) − F ω p,q, ∆ to be a foliation is ofgeneralized curve type. We begin studying what happens when d = gcd( p, q ) = 2.Let us consider S : f = f f and gω = h d( f f ) + f f η. For S i : f i ( x, y ) = 0,we have GSV ( F , S ) = πi (cid:90) ∂ S d( hg ) hg + ( f , f ) . Analogously,
GSV ( F , S ) = πi (cid:90) ∂ S d( hg ) hg + ( f , f ) . We have12 πi (cid:90) ∂ S ∪ ∂ S d( hg ) hg = GSV ( F , S ) + GSV ( F , S ) − f , f ) . Therefore (see [ Br , page 532]), GSV ( F , S ) = GSV ( F , S ) + GSV ( F , S ) − f , f ) . (10)For d = 2 = gcd( p, q ), we have y p − x q = (cid:89) i =1 ( y p − ζ i x q ) , with ζ = 1 . Let S i : f i ( x, y ) = ( y p − ζ i x q ) and γ i ( t ) = ( t p , A i t q ) with A p i = ζ i a parameteri-zation of S i . Then( f , f ) = ord t ( f ( γ ( t ))) = ord t ( t pq (1 − ζ )) = pq . (11)Remember that ω p,q, ∆ = ( − qx q − − qy ∆)d x + ( py p − + px ∆)d y , thus GSV ( F , S ) = ord t (cid:16) t pq ( ζ −
1) + ( ζ − A p − t p + q − pq ∆( t p , A t q ) (cid:17) . (12)If we consider (∆ , y p − x q ) > PH ( p,q ) −
1, from (12) we have that
GSV ( F , S ) = pq .Similarly, it turns out that GSV ( F , S ) = pq .After (11) and (10) we have GSV ( F , S ) = 0, which is equivalent to ω p,q, (cid:52) , so thefoliaction F generalized curve type when d = 2.In general [ Br ], when S : f = f · · · f d , we have to GSV ( F , S ) = d (cid:88) i =1 GSV ( F , S i ) − N (cid:88) i (cid:54) = ji =1 ( f i , f j ) , (13)where N = (cid:18) d (cid:19) , GSV ( F , S i ) = ( d − pqd , and ( f i , f j ) = pqd . Therefore, from(13) we get GSV ( F , S ) = 0 . (14)Hence the following proposition holds. Proposition 5.7.
Let F p,q, ∆ be a non dicritical foliation and suppose that (∆ , y p − x q ) > PH ( p,q ) − . Then F p,q, ∆ is of the generalized curve type. Proof of Theorem 1.3.
It is an immediate consequence of Propositions 5.6 and5.7. (cid:3)
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HARACTERIZATION OF SECOND TYPE PLANE FOLIATIONS USING NEWTON POLYGONS15 (Percy Fern´andez)
Dpto. Ciencias - Secci´on Matem´aticas, Pontificia Universidad Cat´olicadel Per´u, Av. Universitaria 1801, San Miguel, Lima 32, Peru
E-mail address : [email protected] (Evelia R. Garc´ıa Barroso) Dpto. Matem´aticas, Estad´ıstica e Investigaci´on Operativa,Secci´on de Matem´aticas, Universidad de La Laguna. Apartado de Correos 456. 38200La Laguna, Tenerife, Spain.
E-mail address : [email protected] (Nancy Saravia-Molina) Dpto. Ciencias - Secci´on Matem´aticas, Pontificia UniversidadCat´olica del Per´u, Av. Universitaria 1801, San Miguel, Lima 32, Peru
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