Truncated Local Uniformization of Formal Integrable Differential Forms

We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential $1$-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.


Introduction
In this work, we obtain Local Uniformization for rational foliations in any ambient dimension, along a rational valuation of rank one, as well as a Truncated Local Uniformization of formal integrable differential 1-forms.
Let us consider a birational class C of projective varieties over a characteristic zero base field k. That is, we take a field extension K/k, where K is the field of rational functions K = k(M ) of any projective model M ∈ C. A rational foliation F over K/k is any one-dimensional K-vector subspace of the Kähler differentials Ω K/k , whose elements satisfy Frobenius integrability condition ω ∧ dω = 0.
In this paper we prove: Theorem 1. Let F ⊂ Ω K/k be a rational foliation over K/k and consider a rank one k-rational valuation ring R of K. There is a birational projective model M of K such that F is simple at the center P of R in M .
This is a result of Local Uniformization in the sense of Zariski [32], where the objects to be considered are rational foliations. Let us note that the case of a rational function φ ∈ K is included, when we consider the differential ω = dφ; in this case we obtain a classical Local Uniformization of the divisor div(φ).
The reduction of singularities of codimension one foliations is an open problem in dimension bigger of equal than four. We have positive answers by Seidenberg [29] in 1968, for the two-dimensional case, and by Cano [6] in 2004, for the threedimensional case. This is in contrast with Hironaka's results [22] of 1964, that provide a reduction of singularities for algebraic varieties in characteristic zero and any dimension.
We have to deal with the possibility of having nonzero objects with infinite value. This is a reason for making the hard part of the proof in terms of formal objects and in a "truncated way". The infinite value for a nonzero formal function only comes when it is a non rational function, see for instance the works on the implicit ideals [21,17]. However, when we are considering 1-differential forms, the property of having infinite value may appear even with differential forms having polynomial coefficients. This is the case of the well known Euler's Equation in dimension two.
The meaning of "simple integrable 1-differential form" has been established in previous works. In dimension two by Seidenberg [29] and in any dimension by Cerveau-Mattei [13], Mattei-Moussu [24], Mattei [23], Cano-Cerveau [7] and Cano [6] among others. The definition contains the case of the differential of a monomial (normal crossings) and several versions of saddle-nodes.
As it is standard in reduction of singularities, a normal crossings divisor is present in the definition of "final" points after reduction of singularities. It can be an exceptional divisor created along the reduction of singularities process, or also an originally prescribed divisor (for instance, in arguments working by induction). Let us recall the local definition of simple point for the case of a function. A function f is simple with respect to a normal crossings divisor x 1 x 2 · · · x r = 0 if one of the following properties holds: • The function is a monomial in x times a unit U . That is f = U x a . • There is a new local coordinate y and a unit U such that f = U y b x a .
In the first case, that we call the corner case, the zeroes of f are contained in the divisor; in the second case, that we call trace case, the set of zeroes of f contains y = 0. When f is a rational function, we can avoid trace points along the valuation ring R; otherwise, the value of f would be infinity. Nevertheless, it is possible to get trace points for a rational differential 1-form; in the two dimensional case, this indicates the presence of a formal non-rational invariant curve.
The definition of a simple formal integrable differential 1-form ω is compatible with the above one given for functions, in the sense that that if f is not a unit and it is simple, then df will be simple. The precise definition, the existence of normal formal forms and other properties may be found in [5,6,7,20], where "simple points" are "pre-simple points" with a diophantine additional condition, that is automatically satisfied in the case of functions. By a paper of Fernández-Duque [20], it is possible to get only simple singularities when we start with pre-simple ones in any dimension; in other words, we can globally obtain the diophantine condition that makes the difference between pre-simples and simples. This means that it is only necessary to obtain pre-simple points in Theorem 1. We recall that ω is pre-simple with respect to the divisor x 1 x 2 · · · x r = 0, if it can be written in a logarithmic way as ω = f ω * , where b j dy j and one of the following properties holds: a) There is a unit among the coefficients a 1 , a 2 , . . . , a r . b) There is a unit among the coefficients b j or there is a coefficient a i whose linear part is not a linear combination of x 1 , x 2 , . . . , x r .
In case a), we have a pre-simple corner; otherwise, we have a pre-simple trace point.
There are several results concerning the reduction of singularities of codimension one singular foliations and dynamical systems given by vector fields in dimension bigger that two. For vector fields over three dimensional ambient spaces: we have Local Uniformization type results in [5,12], a global reduction of singularities in the real case by Panazzolo [26] and in terms of stacks and orbifolds, by Panazzolo-McQuillan [27]. In general dimension, there are papers of Belotto [3,4] where he performs reduction of singularities of ideals and varieties conditioned to simple foliations. Related problems are the monomialization of morphisms by Cutkosky [16], the reduction of first integrals of dynamical systems [2] or the works of Abramovich [1]. Some of the "extra" technical difficulties in that problems are of the same nature as some of those we encounter in the case of foliations. Surprisingly, this is also true with respect to the known results for the three dimensional case of schemes in positive or mixed characteristic [14,15].
In this paper, we consider k-rational valuations of rank one. In classical Zariski's approach, this is the case where the problem is concentrated: one can pass to the case of a general valuation, following for instance the paper [25]. In the case or rational foliations, there are new difficulties when we consider general valuations, that we plan to solve in a forthcoming paper.
Anyway, the k-rational valuations of rank one have a geometrical and dynamical interpretation in the real case, that has been considered in [11]. The fact of being k-rational means that each time we blow-up, the center of the valuation is a krational point, hence a "true" point. In the real case, we have valuations of this type given by transcendental non oscillating curves, see [9,10,11]. The property of being of rank one means that we cannot decompose the valuation; it may be interpreted by saying that the curve has no flat contact with any hypersurface.
Let us give a few comments on the technical structure of this paper. First of all, as it is ubiquitous in problems of local uniformization, for instance for the positive characteristic case [31], we have to deal with the possibility of having "bad accumulation of values". In zero characteristic, the classical Tschirnhausen transformation is usually the tool that allows us to avoid the accumulation of values. A big part of the paper is devoted to facilitate the use of Tschirnhausen transformations in the case of integrable forms.
Instead of giving a direct proof of Theorem 1, we prove a statement of Truncated Local Uniformization and we derive Theorem 1 from it. This has several advantages. The first one is that we can deal with infinite value objects and in fact the result is valid for formal differential forms. Moreover, the structure of our induction process is simplified, since it is possible to "decompose" a formal series with respect to groups of variables and then we can apply induction to the coefficients.
Let us explain the Truncated Local Uniformization statements. Let Γ be the value group of the valuation ν associated to R. We know that Γ ⊂ R, since we are dealing with a rank one valuation. The truncated statements are relative to a "truncation value" γ ∈ Γ.
Recall that we are always working with respect to a list x = (x 1 , x 2 , . . . , x r ) of parameters that represents the prescribed divisor, that we call the independent parameters. We require the values ν(x i ) to be a Q-basis of Γ⊗ Z Q. The Q-dimension r of Γ ⊗ Z Q is the so-called rational rank of the valuation. We complete x with a list of "dependent parameters" Let us give the definition of truncated γ-final formal functions and truncated γ-final formal differential 1-forms. Given a formal function we define the explicit value ν A (f ) by ν A (f ) = min{ν(x I ); f I (y) = 0}. We say that f is γ-final if one of the following properties holds: • Recessive case: ν A (f ) > γ.
The dominant case is very close to the definiton of corner singularity: in fact, by combinatorial blow-ups, concerning only the indepedent parameters x, we can obtain the additional property that f = x I0 U , where U is a unit. Now, let us consider a formal 1-differential form ω, that we write in a logarithmic way as The explicit value ν A (ω) is defined by ν A (ω) = min{{ν A (f i )} r i=1 , {ν A (g j )} m−r j=1 }. We say that ω is γ-final if one of the following properties holds: • Recessive case: ν A (ω) > γ.
• Dominant case: ν A (ω) ≤ γ and there is a coefficient The proof of the γ-truncated local uniformization goes by induction on the number m−r of dependent variables y. In particular, we need a truncated version of Frobenius integrability condition ω ∧ dω = 0, compatible with the induction procedure. We say that ω satisfies the γ-truncated Frobenius integrability condition if ν A (ω ∧ dω) ≥ 2γ.
Here we see one of the advantages of the case of a formal function f . The differential df trivially satisfies the integrability condition. Moreover, when we decompose f as a power series in the last dependent variable, the coefficients are also formal functions that also satisfies the integrability condition. Note that Section 7 is devoted to the preparation procedure. It should not be necessary for the case of a formal function, since in this case the preparation is a direct consequence of the induction hypothesis.
The Truncated Local Uniformization may be stated as follows: Theorem 2. Consider a nonsingular algebraic variety M over a characteristic zero field k with field of rational functions K = k(M ), and let R be a rank one krational valuation ring of K. Fix a value γ in the value group of R. For any formal differential 1-form ω in the center P of R in M satisfying the γ-truncated Frobenius integrability condition, there is a birational regular morphism π : M ′ → M , that is a composition of blow-ups with non-singular centers, such that the transform π * ω of ω is γ-final at the center P ′ of R in M ′ .
We first give a proof of the "local statement" Theorem 3 in Section 6. To see that Theorem 3 implies Theorem 2, it is enough to assure that the local centers of blow-ups can be made global non-singular ones, just by performing additional blowing-ups external to the centers of the valuation. This is a standard argument of reduction of singularities that we do not detail in the text. Theorem 3 will be a consequence of Theorem 5, that is a stronger inductive version of it.
We devote Sections 6, 7, 8 and 9 to the proof of the Theorem 5. In Section 10, we show how Theorem 5 implies Theorem 1, with arguments directly related to the proof of Theorem 5.
Let us give an idea of the structure of the proof of Theorem 5. We do induction on the number of dependent variables in y that appear in the expression of ω. If no dependent variables appear, we are done, since ω is automatically γ-final. We rename the dependent variables that appear in the expression of ω as (y ≤ℓ , z) = (y 1 , y 2 , . . . , y ℓ , z) and we assume that we have γ-truncated local uniformization when only ℓ dependent variables appear. Then, we make a decomposition of ω as Each ω s is the s-level of ω. Note that we could apply induction to h s and η s , but for this we need to control the explicit value of η s ∧ dη s . Note also that h 0 = 0, since ω has no poles along z = 0; this feature is interesting in some of our preparation arguments. Now, we draw a Newton Polygon N ω ⊂ R 2

≥0
from the cloud of points (s, ν A (ω s )), for s ≥ 0. The importance of the Newton Polygon in the induction step is due to the following remark: "If N ω has the only vertex (ρ, 0) and The objective is to obtain ω with the above property, or such that ν A (ω) > γ, after suitable transformations A → A ′ between parameterized local models. Let us say a word about the transformations we use. They are of three types: • Blow-ups in the independent variables.
• Puiseux's packages. The blow-ups in the independent variables are blow-ups with centers x i = x j = 0 given by two independent variables. They are combinatorial along the valuation; we can use them, for instance, to principalize ideals given by monomials in the variables x.
The nested coordinate changes do not affect to the ambient space. They have the form . These coordinate changes are necessary in order to avoid problems of accumulation of values.
We have already considered Puiseux's packages in previous works [12]. They are related with Perron's transformations, the key polynomials of a valuation and binomial ideals, see for instance [18,31,32]. In dimension two, they are close to the Puiseux's pairs of a plane branch. We can understand them through the rational contact function Φ = z d /x p , which satisfies ν(Φ) = 0. The Puiseux's package can be interpreted as a local uniformization of the hypersurface The number d > 0 is called the ramification index of the Puiseux's package. When d = 1, the above hypersurface is non-singular and thae equations defining the transformation have an appropriate form; in fact, this is the case we encounter at the end of the proof.
Thus, we try to control the evolution of the Newton polygon after performing the above transformations and taking into account the induction hypothesis. We proceed in two steps. First, we perform a γ-preparation of ω. Second, once ω is γ-prepared, we provide a control of the evolution of the critical height χ A (ω), under Puiseux's packages and nested coordinate changes followed by γ-preparations.
Roughly speaking, the γ-preparation of ω consists in obtaining a situation where the relevant levels are ρ-final, with respect to the abscissa ρ determined by the polygon. That is, the "important" part of the level may be read in the coordinates x. In order to obtain the γ-preparation, we need to apply induction to some of the forms η s . The truncated integrability properties of η s do not allow to get a direct γpreparation. We perform first an approximate preparation, thanks to the properties of truncated integrability of the differential parts η s of the levels. We complete the preparation, that we call γ-strict preparation, thanks to some additional properties that are consequence of the hypothesis of truncated integrability and De Rham-Saito type results of truncated division. This part of the proof goes along Sections 6, 7 and 8.
Once we have obtained a γ-preparation, we devote Section 9 to the control of the behaviour of the Newton Polygon under Puiseux's packages and nested coordinate changes, followed by new γ-preparations. Roughly speaking, we do what is necessary in order to obtain a situation modelled on the behaviour of Newton Polygon of plane branches, when we perform a sequence of blow-ups associated to a Puiseux's pair. The shape of Newton Polygon is described by means of numerical invariants. The most important for us is the critical height χ A (ω). The critical segment is the vertex or segment of contact between the Newton Polygon and the lines of slope −1/ν(z). The critical vertex is the highest vertex in the critical segment, and χ A (ω) is the ordinate of the critical vertex.
The new critical height, after a Puiseux's package, is lower or equal that the preceding one. If we are able to obtain strict inequalities, we are done. Thus, We have to look at the cases when the critical height stabilizes at χ = χ A ′ (ω) for any "normalized transformation" (A, ω) → (A ′ , ω). The stabilization implies the presence of resonance conditions, that we call r1, r2a and r2b-υ. The resonance r1 occurs only when χ = 1 and we can show that it happens "at most once". The resonances r2 imply that the ramification index, after the necessary γ-preparations, is equal to one; this is a necessary property for our arguments. In the case when χ = 2, we arrive to show quite directly the existence of a Tschirnhausen coordinate change that allows us to "cross the limit imposed by γ". In the case χ = 1, we follow the same general ideas, but it is necessary for us to use truncated cohomological results, namely a generalized and truncated version of Poincaré's Lemma that we include in Section 4.
Acknowledgements: The authors are grateful with O. Piltant, M. Spivakovsky and B. Teissier for many fruitful conversations on the subject. This work has been supported by the Spanish Research Project MTM2016-77642-C2-1-P.

Formal Differential Forms in the Center of a Valuation
Let K/k be a field of rational functions over a base field k of characteristic zero. Along all this paper, we consider a rational k-valuation ring k ⊂ R ⊂ K of rank one. That is, the following properties hold: • The natural mapping k → κ over the residual field κ of R is an isomorphism.
In particular, for any birational model M of K the center of R in M is a k-rational point of M . • The value group Γ = K * /R * of the valuation ν associated to R is isomorphic to a subgroup of (R, +). Once for all, we fix an immersion Γ ⊂ R.
where O A is a regular local ring and (x, y) = (x 1 , x 2 , . . . , x r , y 1 , y 2 , . . . , y m−r ) is a regular system of parameters of O A satisfying the following properties: The parameters x are the independent parameters and y are the dependent parameters of A.

Let us consider a locally parameterized model
Kähler differentials, that is the logarithmic Kähler differentials with respect to the monomial x 1 x 2 · · · x r . We recall that OA/k and both are free modules of finite rank m = dim M , see for instance [19]. Let Ω 1 where O A is the completion of O A with respect to its maximal ideal m A . The elements of Ω 1 A are called A-formal logarithmic differential 1-forms or simply formal logarithmic differential 1-forms, it the reference to A is obvious.
and Ω 1 A depend only on the ideal (x 1 ) generated by the monomial We have a well defined exterior derivative d : Ω p A −→ Ω p+1 A , as well as an exterior product α, β → α ∧ β in Ω • A . Remark 2.3. Note that the valuation ν is defined for the elements f ∈ K \ {0}.

Explicit Values and Final Truncated Differential Forms
Let us consider a parameterized local model A = (O A ; x, y).

Explicit Values.
Given δ ∈ R ≥0 , we define the ideals I δ A and I δ+ We obtain a filtration of O A that we call the explicit filtration. It depends only on the ideal (x 1 ). In the same way, the explicit filtration of O A = Ω 0 A is given by the family of ideals It is a well ordered subset of R ≥0 . In particular, for any δ ∈ R ≥0 there are unique δ 0 , δ 1 ∈ V A with δ ≤ δ 0 ≤ δ 1 such that Thus the family of ideals {I δ A ; δ ∈ V A } gives the explicit filtration. Let us also note that for any y]] stands for the formal series ring in the variables x, y with coefficients in k. Write f ∈ Ω 0 A as a formal series We have that ν A (f ) = min{ν(x I ); A -module of finite rank. We extend the explicit filtration to N by considering the family of submodules I δ A N and I δ+ A does not depend on the choice of the basis. From now on, we denote by ν A the explicit order in the free Ω 0 A -modules Ω p A . We have the following standard valuative properties:

3) For any α ∈ Ω p
A and β ∈ Ω q A we have ν A (α ∧ β) ≥ ν A (α) + ν A (β). Next, we give the definition of the main technical objects in this paper: Definition 3.2. Let A be a locally parameterized model. Consider ω ∈ Ω 1 A and a real number γ ∈ R. We say that (A, ω) is a γ-truncated formal foliated space if and only if ω satisfies the γ-truncated integrability condition A γ-truncated formal foliated space (A, ω) is also γ ′ -truncated for any γ ′ ≤ γ. If ω satisfies Frobenius integrability condition ω ∧dω = 0, then (A, ω) is a γ-truncated foliated space, for any γ ∈ R; in this case, we say that (A, ω) is a formal foliated space. In particular, we have a formal foliated space (A, df ) associated to a given formal function f ∈ Ω 0 A .
3.2. Final truncated differential forms. For any δ ∈ V A , let G δ A denote the quotient G δ The graded algebra G A = ⊕ δ G δ A , associated to the explicit filtration of Ω 0 A , can be identified with a weighted polynomial algebra of G A is a complete regular local ring with maximal ideal m A = m A /(x) and residual field naturally isomorphic to k. In particular, we have that Let us introduce the definition of final formal functions. Definition 3.3. Consider a real number γ ∈ R and a formal function f ∈ Ω 0 A . We say that (A, f ) is γ-final recessive if δ > γ, where δ = ν A (f ). We say that (A, f ) is γ-final dominant if δ ≤ γ and we have that We say that Let us introduce some notations to facilitate the generalization of the above Definition 3.3 to higher order forms. We denote by C 0 y]]. Let δ denote the value ν A (f ). We can write f as Consider an element ω ∈ Ω 1 A and write it as under the natural morphism in Equation (3.1). We extend Definition 3.3 to formal differential 1-forms ω ∈ Ω 1 A as follows:

Explicit values under Differentiation.
We compare here the explicit value of dα ∈ Ω p+1 A with the explicit value of α ∈ Ω p A .
Then f = ν(x I )≥δ x I f I and ∂f = ν(x I )≥δ x I g I , where g I = ∂f I +f I r j=1 i j a j and I = (i 1 , i 2 , . . . , i r ). Thus, we have that ν A (∂f ) ≥ δ.
In the case that f ∈ m A Ω 0 A , then ν A (df ) = ν A (f ). More generally, for any α ∈ Ω p A we have that ν A (dα) ≥ ν A (α).
We can write Then c 0 A (f ) = 0 is equivalent to saying that λ = 0, where f I = λ + |J|>0 f I,J y J . On the other hand, we have df = f I d(x I ) + x I df I + df . Since ν A (df ) > δ and d(x I ) = x I r j=1 i j dx j /x j with I = 0, we have that c 1 M (df ) = 0 is also equivalent to λ = 0.

Truncated Cohomological Statements
Next result is a truncated version of De Rham-Saito division [28].
Since α is 0-final dominant, there is at least an unit between the coefficients a 1 , a 2 , . . . , a r . Without lost of generality, we can assume that a 1 is a unit. Since So, there are functionsf i with ν A (f i ) ≥ ρ such that f i = Ha i +f i for i = 2, 3, . . . , r.
Let us consider a "value of truncation" ρ ∈ R ∪ {+∞}, where the case ρ = +∞ means "no truncation". In Proposition 4.2 below we state and prove a truncated µ-multivaluated and logarithmic generalization of formal Poincaré's Lemma.
We know that d I+µ (η I ) = 0 for any I such that ν(x I ) < ρ. Let us consider one such I and put ρ = I + µ. Let us split η I =η I + η * I wherẽ implies that dη * I = 0, since the coefficients of d ρ η I for dy j ∧ dy s correspond exactly to the coefficients of dη * I . By applying the standard formal Poincaré's Lemma, we find F * ∈ k[[y]] such that dF * = η * I . Now, we have two cases to consider: Case ρ = 0: By Equation (4.1), we deduce that dx ρ x ρ ∧η I = 0, just by looking to the coefficients of d ρ η I of the terms in (dx i /x i ) ∧ (dx ℓ /x ℓ ). Since ρ = 0,the above proportionality implies that there is f I ∈ k[[y]] such that Finally, in this case we have that η I = d ρ (f I ) and hence x I η I = d µ (x I f I ). Case µ = −I: We have that dη I = dη I + dη * I = 0. Recalling that dη * I = 0, we conclude that dη I = 0. This implies that df i = 0 for any i = 1, 2, . . . , r and hence f i = λ i ∈ k for any i = 1, 2, . . . , r. That is where f I = F * in this case. More precisely, we have where we recall that µ = −I. Finally, let us put f = νA(x I )<ρ x I f I . We have . This ends the proof.
By Proposition 4.2, the fact that dα = 0 implies that there is h ∈ Ω 0 A and µ ∈ C r such that We call dx µ /x µ the residual part of α in A. Take an index i 0 ∈ {1, 2, . . . , r} such that µ i0 = 0 and consider the unit We have Corollary 4.1. Let α ∈ Ω 1 A be 0-final dominant with dα = 0 and residual part dx µ /x µ . Let η ∈ Ω 1 A be such that ν A (d α η) ≥ ρ. We can decompose η as η = θ +η, where ν A (η) ≥ ρ and there is a formal function F ∈ Ω 0 A such that θ has the form , for a formal unit W and x i = x i for i = i 0 . Moreover, in the case that −µ / ∈ Z ≥0 we have λ i = 0, for i = 1, 2, . . . , r. In particular, we have θ ∧ dθ = 0.
Proof. Follows from Proposition 4.2, noting that α = dx µ /x µ . That is, the derivative d α is expressed in the new independent coordinatesx as dμ and moreover, we recall that ν A is independent of the coordinate change x →x.

Transformations of Locally Parameterized Models
The allowed transformations between locally parameterized models are finite compositions of the following types of elementary allowed transformations: • ℓ-coordinate changes.
• Independent blow-ups of locally parameterized models.

We obtain a new locally parameterized model
where A → A ′ is an ℓ-blow-up with translation and A → A is a finite composition of independent blow-ups and ℓ-combinatorial blow-ups. Let us define these types of blow-ups: 1) An ℓ-combinatorial blow-up A → A * is given by the choice of an x i such that ν(y ℓ ) = ν(x i ). As before, we take a projective model M associated to A and the blow-up π : To obtain the regular system of parameters x * , y * we have two cases: We take a projective model M associated to A and the blow-up π : We obtain the regular system of parameters x * , y * as follows.
Remark 5.1. Let us note that an ℓ-combinatorial blow-up is not an allowed transformation itself. Nevertheless, an ℓ-blow-up with translation defines an ℓ-Puiseux's package.
The existence of at least one ℓ-Puiseux's package A → A ′ is proved in [12]. It is related with the so called ℓ-contact rational function Φ of A. Since the values of the independent parameters define a basis for Γ ⊗ Z Q, there are unique integers d > 0 and p = (p 1 , p 2 , . . . , p r ) ∈ Z r such that d, p 1 , p 2 , . . . , p r are coprime and We put Φ = y d ℓ /x p . The number d ≥ 1 is called the ℓ-ramification index of A. There is a unique 0 = λ ∈ k such that ν(Φ − λ) > 0. Note that, in the case of a blow-up with translation, the ℓ-contact rational function is given by y ℓ /x i .

Equations for Puiseux's Packages
In this subsection we recall formulas in [12] relating the parameters x, y, x, y and x ′ , y ′ .
The relationship between x, y and x ′ , y ′ is given by The relationship between x, y and x, y is given by a (r+1)×(r+1) matrix B = (b ij ), with nonnegative integer coefficients and det B = 1, such that where c si0 = b si0 +b s,r+1 for any s = 1, 2, . . . , r+1 and c ji = b ji , if i = i 0 . Note that we also have that det C = 1 and the coefficients c ji of C are nonnegative integer numbers.
Remark 5.2. We have the following properties: Then det C 0 = 0 since it gives a base change matrix for the bases We have ν(y ℓ ) > 0 and hence y ℓ is not a unit in O A ′ . More precisely, we have that ν A ′ (y d ℓ ) = ν(x p ) > 0 and thus ν A ′ (y ℓ ) > 0. As a consequence of Proposition 3.1, we also have that ν A ′ (dy ℓ ) = ν A ′ (y ℓ ) > 0. 4) The case d = 1 is relevant for our computations. The following properties are equivalent i) d = 1.
ii) The ℓ-combinatorial blow-ups in A → A are all of transversal type.
iii) The matrix C has the form 5) The relationship between the differential forms are given by: where we remark that dΦ/Φ = (1/(y ′ ℓ + λ))dy ′ ℓ .

Stability
Results. The use of allowed transformations in the Truncated Local Uniformization is justified by the results in this subsection. We show that the critical values are not decreasing under allowed transformations and that the stabilization of the critical value characterize final situations.
Remark 5.3. Any allowed transformation A → A ′ gives and injective morphism of local rings O A → O A ′ and also an injective morphism of complete local rings that are compatible with the sum, the exterior product and the exterior differentiation. In order to simplify notations along the paper, when we have ω ∈ Ω p A , we write ω ∈ Ω p A ′ to denote the transform of ω by the considered allowed transformation.
Proof. This result is a direct consequence of the description, given in Subsections 5.1 and 5.2, of the elementary allowed transformations.
is an ℓ-coordinate change, we are done. Thus, we have only to solve the cases of an independent blow-up or an ℓ-Puiseux's package.
Put δ = ν A (α). Let us prove that ν A ′ (α) ≥ δ. By the properties of the explicit order, we have only to consider the case α = f ∈ Ω 0 A . Write f as a finite sum f = ν(x I )≥δ Assume now that α ∈ Ω 1 A . Saying that ν A (α) = δ and c 1 A (α) = 0 is equivalent to saying that if α = x I β +α, ν A (α) > δ and ν(x I ) = δ, then ν A (β) = 0 and c 1 A (β) = 0. Write α = x I β +α as before. By Lemma 5.1 and the previous result, the only thing we have to prove is that We know that there is a unit among the coefficients f s . In the case of an independent blow-up, we have . . , f r )A and A is a matrix with det A = 1 and nonnegative integer coefficients. Then, there is a unit between the coefficients f ′ s and hence ν E ′ (β) = δ and c 1 A ′ (β) = 0. In the case of an ℓ-Puiseux's package, let us write

Consider a sequence of allowed transformations
Proof. Put δ = ν A (α), note that δ < ∞, since α = 0. Let us write α as where, in addition, we write α I as follows: Remark 5.4. Let us fix a real number γ ≥ 0, a locally parameterized model A and 0 = α ∈ Ω p A , p ∈ {0, 1}. In order to obtain a γ-truncated local uniformization of α, a rough idea is to increase the explicit value ν A (α) while α is not final dominant. By Proposition 5.2, we can increase the value, but we have to deal with the possibility of an accumulation of the explicit values before arriving to the truncation limit given by γ. This is one of the classical difficulties in Local Uniformization problems.

Final forms and Independent Blow-ups.
It is a classical result [30,22,32,16] that any monomial ideal in the independent variables becomes a principal ideal under a suitable sequence of independent blow-ups. Let us state here these results in a useful way for the proof of the truncated local uniformization of 1-forms.
There is a transformation A → A ′ obtained as a composition of finitely many independent blow-ups such that the transformed list has the property that x ′ I ′ 0 divides any other m ′ I . Proof. (See the above references). The proof may be done by succesive elimination of vertices of the Newton Polyhedron of L. We leave the details to the reader.
A . There is an allowed transformation A → A ′ , obtained as a composition of finitely many independent blow-ups, such that ω has the form

Truncated Local Uniformization
In Theorem 3 we state the Truncated Local Uniformization for 1-Forms: We get also a Truncated Local Uniformization for Formal Functions as follows: Theorem 4. Consider a formal function f ∈ Ω 0 A and a real number γ ∈ R. There is an allowed transformation Theorem 4 may be considered as an avatar of the classical Zariski's Local Uniformization in [32]. Note that Theorem 4 is a consequence of our main result Theorem 3, when we consider the 1-form ω = df . 6.1. Induction Structure. The proof of Theorem 3 goes by induction on the number I A (ω) of dependent variables involved in the formal 1-form ω. To be precise, satisfies that . . , y s ]], for i = 1, 2, . . . , r, j = 1, 2, . . . , m − r.
Definition 6.1. An allowed transformation A → A ′ is called a ℓ-nested transformation when it is a finite composition of independent blow-ups, ℓ ′ -coordinate changes and ℓ ′ -Puiseux's packages, with ℓ ′ ≤ ℓ.
The inductive version of Theorem 3 that we are going to prove is the following one:

Starting Situation for the Inductive
Step. Next sections are devoted to the proof of Theorem 5 by induction on I A (ω). This is the main technical part in this paper. The proof is divided in two parts. The first one is the preparation procedure. In the second part, we provide a control of the critical height, in a prepared situation, under "normalized" Puiseux's packages and coordinate changes. The rest of this paper, except for the last section, is devoted to the inductive proof of Theorem 5. By Remark 6.1, we know that Theorem 5 is true when we have I A (ω) = 0. Thus, we supose that I A (ω) = ℓ + 1 ≥ 1 and we take the induction hypothesis that says that Theorem 5 is true when I A (ω) ≤ ℓ.
The cloud of points Cl A (ω) is defined by . It has finitely many vertices, each one belonging to the cloud of points. The main vertex is the vertex with highest ordinate, it is also the vertex with minimal abscissa. We call main height to the ordinate of the main vertex. Let us note that the abscissa of the main vertex is equal to ν A (ω). Proposition 6.1 below highlights the inductive role of the Newton Polygon: ) and is called the critical vertex.

Statements of Strict Preparation
We introduce here the definitions and first results concerning the preparation process of the γ-truncated formal foliated space (A, ω). Definition 7.1. Given a real number δ ∈ R we say that the s-level ω s of (A, ω) is Remark 7.1. Let A → A ′ be an ℓ-nested transformation. Then the decomposition into levels given in Equation (6.1) is valid both for (A, ω) and (A ′ , ω). Applying Proposition 5.1 to each level, we have that If ω s is λ s -dominant for each vertex (λ s , s), we have the equality N A ′ (ω) = N A (ω).
is also γ-prepared. Moreover, we have same critical value and same critical segment, that is ς A ′ (ω) = ς A (ω) and C A ′ (ω) = C A (ω). This is a consequence of Proposition 5.1.

7.1.
Maximally Dominant Truncated Foliated Spaces. The first step of the preparation process is to obtain a "maximally dominant polygon". To be precise, we say that (A, ω) is ρ-maximally dominant if and only if, for any s ≥ 0, one of the following properties holds: In view of Propositions 5.1 and 5.2 and Remark 7.1 the property for an s-level of being (ρ − sν(z))-dominant is stable under any ℓ-nested transformation A → A ′ . In particular, the property of being ρ-maximally dominant is stable under further ℓ-nested transformations.
The "dominant preparation" is given by Proposition 7.1 below: Proof. We have only to consider s-levels with 0 ≤ s ≤ γ/ν(z). This is a finite set. Then, it is enough to consider an ℓ-nested transformation that produces the maximum number of (γ − sν(z))-dominant levels.

Pseudo-Prepared and Strictly Prepared Situations. We say that (A, ω)
is γ-pseudo prepared if and only if the following conditions hold: • For any s ≥ 0 we have that h s is (γ − sν(z))-final.
• The 0-level η 0 is γ-final. In view of Proposition 7.1 and using the induction hypothesis applied to formal functions and to η 0 , there is an Definition 7.3. We say that (A, ω) is strictly γ-prepared if and only if it is both γ-prepared and γ-pseudo prepared.
Remark 7.5. Although we only need a result of preparation for our purposes, we shall give a proof of the existence of strict γ-preparation.

Preparation Theorem.
Next Preparation Theorem 6 is the first step in inductive proof of Theorem 5.
The if A → A ′ is an ℓ-nested transformation as in Theorem 6, we say that (A, ω) → (A ′ , ω) is a strict γ-preparation of (A, ω).
Let us note that, in order to prove Theorem 6, we can assume that (A, ω) is γ-pseudo prepared. The next section is devoted to the proof of Theorem 6.

Preparation Process
In this section we give a proof of Theorem 6. In other words, we show the existence of a strict γ-preparation for a given γ-truncated formal foliated space (A, ω), that we suppose to be γ-pseudo prepared without loss of generality.
The γ-preparation will be done in two steps. First, we show that we can approximate the Newton Polygon to the dominant Newton Polygon. Second, we use this approximation to obtain the preparation.
Remark 8.1. The preparation process is done under the induction hypothesis. More precisely, we are going to apply induction hypothesis to the levels in particular, to the 1-forms η s and to the "horizontal coefficients" h s . The induction hypothesis may be directly applied to the formal functions h s , since we do not need to assure any additional property of truncated integrability (recall that dh s is Frobenius integrable, since d(dh s ) = 0). Nevertheless, when we consider the 1-forms η s , we can only apply the induction hypothesis with respect to truncation values ρ for which η s satisfy the ρ-truncated integrability condition. The "previous" approximated preparation is necessary due to this observation.
We obtain the following objects from the dominant cloud of points: is the ordinate of the highest vertex in the γ-dominant critical segment. This vertex has the form ) and is called the γ-dominant critical vertex.
The totally recessive case Cl γ A (ω) = ∅ corresponds to the property ς γ A (ω) = ∞. The dominant case Cl γ A (ω) = ∅ corresponds to the property ς γ A (ω) ≤ γ. Let us also recall that we always have the property ς A (ω) ≤ ς γ A (ω). Then (A, ω) is γ-prepared if and only if either ς A (ω) > γ or the following statements hold: . In this case, we have that C γ A (ω) = C A (ω) and the levels in the cloud of points contributing to the critical segment are only dominant levels. In particular the critical vertex corresponds to a dominant level.
Proof. If ς A (ω) > γ we see that (A, ω) is γ-prepared by definition. Thus, we assume This shows a) and b). Conversely, assume a) and b). For any s-level ω s we have that Corollary 8.1. Assume that (A, ω) is γ-maximally dominant and that one of the following two conditions holds Proof. If ς A (ω) > γ, we are done. Assume that ς γ A (ω) = ∞ and ς A (ω) = γ. Applying Proposition 5.2 we obtain the situation where ς A (ω) > γ and we are done.
If ς γ A (ω) < ∞, we have ς A (ω) = ς γ A (ω) ≤ γ. We can also apply Proposition 5.2 to the non dominant levels ω s such that (ν A (zω s ), s) ∈ L, where L = L γ A (ω) = L A (ω). In this way we obtain Property b) in Proposition 8.1. Proposition 8.1 above and Corollary 8.1 show that we need to "approximate" the critical segment to the γ-dominant critical segment in order to obtain a prepared situation.
, for any s. We have We conclude that This contradicts the fact that µ t+1 ≥ 0.
Let us denote θ δ,ρ (ǫ) = min{1, 2ǫ/(h + 1)(h + 2)}. We introduce now the kind of operations that we perform in the approximate preparation process. Consider two positively convex polygons N, N ′ ⊂ R 2 ≥0 whose vertices have integer ordinates. Given and integer s ≥ 0, we say that N ′ is an s-planning of N if and only if N ′ is contained in the positive convex hull of where Vert(N ) denotes the set of vertices of N . We say that N ′ has been obtained from N by a (δ, ρ, ǫ)-planning operation if and only if N ′ is an s-planning of N , where α N (s) ≥ θ δ,ρ (ǫ) and λ N (s) < ρ − sδ.
Corollary 8.2. Take δ, ρ, ǫ as above and a positively convex polygon N ⊂ R ≥0 whose vertices have integer ordinates. Given any sequence of (δ, ρ, ǫ)-planning operations we have that either Proof. Direct consequence of Lemma 8.2.
Proof. Given a polygon N , let us consider the numbers β N (s) defined by We know that β N (s) = 0 for s < 0 and s ≥ h, where h is the first integer number greater or equal than δ/ρ. If N N ′ is a (δ, ρ, ǫ)-planning operation, we have that β N ′ (s) ≤ β N (s) for any s and moreover, there is an index s 0 such that β N (s 0 ) > 0 and The level s 0 corresponds to the considered vertex in the operation N N ′ . With these properties, it is enough to take b δ,ρ (ǫ) ≥ (h + 1)(ρ + 1)/θ δ,ρ (ǫ).
Proof. In order to simplify notation, let us denote N = N A (ω), δ = ν(z), ρ = min{γ, ς γ A (ω) − ǫ} and λ s = λ N (s) ≤ ν A (zω s ). Let us note that if ς γ A (ω) = ∞, then ρ = γ and if ς γ Let us do an argument by contradiction, assuming that there is no ℓ-nested Up to performing a suitable ℓ-nested transformation, the following properties hold: a) For any level s such that λ s ≤ ρ − sδ there is no new nested transformation with λ ′ s > ρ − sδ. That is, we assume that we have the minimum possible number of levels s with λ s ≤ ρ − sδ. b) For any level s, we have that λ s = ρ − sδ. If we have a level s with λ s = ρ − sδ, it is not a dominant level and we can apply Proposition 5.2 to obtain λ ′ s > γ − sδ in contradiction with the minimality given in a). c) λ 0 > ρ. Recall that η 0 = ω 0 is γ-final. The above properties a),b) and c) are still true under any further ℓ-nested transformation. Let us chooseρ such that ρ <ρ < ς γ A (ω) such that for any s with λ s > ρ − sδ we have λ s >ρ − sδ. Note that this property is stable under any further ℓ-nested transformation. In view of the above reductions of the problem, we have the following properties under any ℓ-nested transformation A → A ′ : i) λ 0 >ρ and λ ′ 0 >ρ. ii) For any s such that λ s ≥ ρ − sδ, then λ s >ρ − sδ and λ ′ s >ρ − sδ. iii) For any s such that λ s ≤ ρ − sδ, then λ s < ρ − sδ and λ ′ s < ρ − sδ. Let us consider now the following "Planning Property": (P): For any s with λ s ≤ ρ − sδ, there is an ℓ-nested transformation A → A ′ such that the transformed Newton Polygon N ′ is an splanning of N . If (P) is true, we end as follows. Take ε =ρ − ρ. By Lemma 8.2 we can perform a (γ, δ, ε)-transformation N ′ of N induced by an ℓ-nested transformation A → A ′ . The transformed polygon N ′ is still in the same situation in view of properties i), ii) and iii). We repeat indefinitely the operation. This contradicts Lemma 8.3. Now, let us show that Property (P) holds. Take s such that λ s ≤ ρ − sδ, note that s ≥ 1. For any j ≥ 1 we have that Let us recall that ν A (Θ 2s ) ≥ 2γ, see Equation (8.2). We deduce that where 2̺ = min{2γ, λ s+1 + λ s−1 }. On the other hand, let us remark that By a new application of the induction hypothesis in view of Equation (8.3), we can perform an ℓ-nested transformation A → A ′ such that (A ′ , η s ) is ̺-final. Moreover, since ̺ ≤ ρ − sδ we have that (A ′ , h s ) is also ̺-final and hence (A ′ , ω s ) is also ̺-final. We know that (A, ω s ) is not ̺-final dominant, since otherwise it would be (ρ − sδ)-final dominant and ρ < ς γ A (ω). We conclude that λ ′ s ≥ (λ s+1 + λ s−1 )/2 and thus N ′ has been obtained by an s-planning of N . Corollary 8.3 (Approximate Preparation). Let us assume that (A, ω) is γ-pseudo prepared. We have the following properties: Preparation. We complete here the proof of Theorem 6.
We follow an argument by contradiction, assuming that Theorem 6 does not hold for a given (A, ω). Denote by s 0 ≤ s 1 the levels corresponding to the vertices of the dominant critical segment and by [s 0 , s 1 ] = {s; s 0 ≤ s ≤ s 1 }. There is a positive µ > 0 such that after a suitable ℓ-nested transformation, the following holds: 2) For any s-level such that ν A (zω s ) < ς − sδ and any ℓ-nested transformation A → A ′ , we have that ν A ′ (zω s ) < ς − sδ.
Moreover, in view of Corollary 8.3, given ǫ > 0 we can perform an ℓ-nested transformation such that ν A (zω s ) > ς − sδ − ǫ for any s and this property is stable under any further ℓ-nested transformation. Let us note that our contradiction hypothesis states that there is at least one level ω s such that ν A (zω s ) < ς − sδ. Lets denote the minimum of the indices s such that ν A (zω s ) < ς − sδ. We remark thats ≥ 1 since the level ω 0 = η 0 is γ-final. We consider the two main situationss < s 0 ands > s 0 that we call respectively the recessive case and the dominant case.
By induction hypothesis we can perform an ℓ-nested transformation in such a way that the levels should be (ς −sδ)-final. Sinces < s 0 , we obtain ν A (ωs) > ς − sδ, this is the desired contradiction. Now, consider the dominant cases > s 0 . For any j < s 0 we have Moreover, for any s 0 ≤ j ≤s we have that ν A (h j ) ≥ ς − jδ. Otherwise, by performing an ℓ-nested transformation we can obtain that ν A (zω j ) > ς − jδ − ǫ > ν A (h j ), contradicting the definition of the dominant abscissa. Noting that ν(zω j ) ≥ ς − jδ for any j <s, we conclude from Equation (8.5) that Now, we are going to consider separately the cases where ν A (h s0 ) > ς − s 0 δ and ν A (h s0 ) = ς − s 0 δ.

Control by the Critical Height
In this section we end the proof of Theorem 5. We take the assumptions and notations as in Section 6. Thus, we fix ℓ ≥ 0, we assume the induction hypothesis, that is, Theorem 5 is true for 1-forms η with I A (η) ≤ ℓ. We consider γ-truncated formal foliated space (A, ω) with I A (ω) = ℓ + 1 and we intend to show the existence of a (ℓ + 1)-nested transformation A → B such that (B, ω) is γ-final.
By Preparation Theorem 6, we start with a strictly γ-prepared (A, ω). The proof of Theorem 5 follows from the control of the critical value ς A (ω) and the critical height χ A (ω) under a type of (ℓ + 1)-nested transformations that we call normalized transformations.

Control of the Critical Height and Critical
Value. Let us state here the results we prove in next subsections, in order to prove Theorem 5. We start with a strictly γ-prepared (A, ω). Because of the results in Propositions 9.1, 9.2 and 9.3, we see that Theorem 5 is a consequence of Propositions 9.4 and 9.5 below: Proposition 9.4. Let us consider an integer number χ ≥ 2. There is no strictly γ-prepared (A, ω) with the property of χ-fixed critical height. Proposition 9.5. There is no strictly γ-prepared (A, ω) with the property of 1-fixed critical height.
When we have ς A ⋆ (ω) ≤ γ and χ A ⋆ (ω) = χ A (ω) ≥ 1 under a normalized Puiseux's package, we see that (A, ω) satisfies certain resonance conditions r1 or r2. Condition r1 occurs "at most once" and only when χ A (ω) = 1. On the other hand, condition r2 implies that the ramification index of A is 1. In this way, we we obtain situations where it is possible to perform Tschirnhausen transformations to "escape" from a situation of χ-fixed critical height. 9.3. Reduced Part of a Level. We consider here a γ-truncated formal foliated space (A, ω), not necessarily strictly γ-prepared. The reduced partω A s of the s-level ω s of (A, ω) appears in many of our computations.
Let us recall the definition of the abscissa λ A,ω (s) given in Subsection 8.3: Note that λ A,ω (s) = ς A (ω) − sν(z), when s corresponds to a level in the critical segment. If ν A (zω s ) > λ A,ω (s), we writeω A s = 0. Assume ν A (zω s ) = λ A,ω (s) and write Writē The reduced partω A s of the level ω s is defined byω A s =ω * s . Let us precise the nature ofω A s . If we consider the k-vector space Ω 1 A defined by A ; we will not insist on this formalism. Anyway, we have an isomorphism of k-vector spaces Ω First, let us give some elementary remarks on positively convex polygons N ⊂ R 2 ≥0 given by a cloud of points in R ≥0 × Z ≥0 , se also Subsection 8.3. For any δ > 0, let us denote ς δ (N ) = min{α + δβ; (α, β) ∈ N } = max{ρ; N ⊂ H + δ (ρ)}. The δ-critical vertex is the highest vertex of N such that α + δβ = ς δ (N ) and the δ-critical height χ δ (N ) is the ordinate of the δ-critical vertex.
≥0 be two positively convex polygons with vertices in (R) ≥0 × Z ≥0 . Let us consider δ 0 > 0. The following statements are equivalent: Proof. It is a standard verification on positively convex polygons. Proposition 9.6. Consider a γ-truncated foliated space (A, ω) and let A → A ′ be a (ℓ + 1)-coordinate change.
As a consequence, we have
Let us note that for t > s we have that λ s − λ t < (t − s)δ 0 . By Equations (9.2), we conclude that This shows thatω A s =ω A ′ s , for any s ≥ χ. We also get that vec A Next result proves Proposition 9.3 in the case of a normalized coordinate change: Proof. Let A → A ′ be the (ℓ + 1)-coordinate change that we follow by a strict γ-preparation (A ′ , ω) → (A ⋆ , ω). In view of Proposition 9.6 and by Remark 9.4, the critical vertex of (A, ω) is also a dominant vertex of N A ′ (ω). Since ν(z ′ ) ≥ ν(z), this vertex is preserved under the strict γ-preparation, and it is higher or equal than the new critical height. 9.5. The Critical Height under a Puiseux's package. Let us take a strictly γ-prepared (A, ω) and consider a normalized Puiseux's package We recall that it is composed of an (ℓ + 1)-Puiseux's package A → A ′ followed by a strict γ-preparation (A ′ , ω) → (A ⋆ , ω).
Let us denote by Φ = z d /x p the contact rational function for the Puiseux's package, where the number "d" is the ramification index of the Puiseux's package. We recall that ν(Φ) = 0 and that Φ = z ′ + λ, where 0 = λ ∈ k is uniquely determined.
Let us recall the level decomposition ω = s z s ω s given in Equation (6.1).

Proof. The stability results under a γ-strict preparation show that ν
. This ends the proof.
Remark 9.6. We obtain Proposition 9.1 as a consequence of Proposition 9.7, just by noting that Let us consider a number ρ lower or equal than the main abscissa ν A (ω). We define the ρ-dominant main height ρ A (ω) by ρ A (ω) = min{s; ν A (zω s ) = ρ andω A s = 0}. In the rest of this subsection, we put ς = ς A (ω) and we assume that ς ≤ γ.
Proof. Let us note that ς ≤ ν A ′ (ω) in view of Proposition 9.7. Then ς A ′ (ω) makes sense. Assume that ς A ⋆ (ω) ≤ γ and put ′ = ς A ′ (ω), that we suppose ′ < ∞. In particular, we have that ς = ν A ′ (ω) is the main abscissa of N A (ω). We have two possible situations: In the process of γ-dominant preparation, see Section 7.1 and Proposition 7.1, we have only to consider levels strictly under . Since we assume that ς A ⋆ (ω) ≤ γ, the critical vertex of (A ⋆ , ω) corresponds to one of that levels. Then In this case, the point (ς, ′ ) of the Newton Puiseux's polygon is persistent under the process of strict preparation A ′ → A ⋆ as well as the main abscissa ς. Then This ends the proof.
Remark 9.7. In the above proof, the only possibility to have χ A ⋆ (ω) = ′ is that (ς, ′ ) is both the main and the critical vertex of N A ⋆ (ω).
Lemma 9.3. We have ς A ′ (ω) ≤ χ A (ω). Remark 9.8. We obtain Proposition 9.3 for the case of a normalized Puiseux's package as a corollary of Lemmas 9.3 and 9.2. We also obtain Proposition 9.2, as follows: the fact that ς A ′ (ω) = 0 implies that ς is the main abscissa and (ς, 0) is the main vertex that corresponds to a ς-final level. In view of Proposition 6.1 we have that (A ′ , ω) is γ-final and thus (A ⋆ , ω) also is γ-final.
Let us start the proof of Lemma 9.3. Denote δ = ν(z) and ′ = ς A ′ (ω). Let s 0 and χ, with s 0 ≤ χ be the ordinates of the vertices of the dominant critical segment C A (ω). We recall that χ = χ A (ω) is the critical height of (A, ω). The case χ = 0 is straighforward and we leave it to the reader. Thus we assume χ ≥ 1.
Since (A, ω) is strictly γ-prepared, we can decompose ω = ω * +ω where ς A (ω) > ς and ω * may be expressed as follows ] and the following properties hold: Each ω * s with ω * s = 0 is 0-final with respect to A and more precisely we have: a) If η * s = 0 there is a unit among the coefficients f * i,s , for i = 1, 2, . . . , r. b) If h * s = 0 then h * s is a unit in k[[y ℓ ]]. Let us denote µ s = h * s (0), the class of h * s modulo the maximal ideal. In the same way, we denote λ s,i = f * s,i (0). Now, we decompose ω * s =ω * s +ω * s , wherē We have that ω * s = 0 if and only ifω * s = 0. Note thatω * s is not 0-final dominant and we can write it as y j ω * s,j + g * s,j dy j . (9.5) Let us writeω * = χ s=s0 z s x Isω * s andω * = χ s=s0 z s x Isω * s . Hence ω * =ω * +ω * . Definition 9.3. We callω * the reduced critical part of ω with respect to A.
i for all i ≥ 0. Let us remark that any element θ ∈ V A ′ [Φ] may be written in a unique way as a finite sum Next lemma provides a way to compute 0 We have Proof. It is enough to show thatω * s ∈ V A ′ , for any s 0 ≤ s ≤ χ. Writeω * s as in Equation (9.4). Recalling the matrix C in Equation (5.3), we havē where (λ ′ s , µ ′ s ) = (λ s , µ s )C. Proposition 9.8. We have 0 A ′ (ᾱ) ≤ χ. Moreover, if 0 A ′ (ᾱ) = χ one of the following two conditions "r1" or "r2" holds: andᾱ has one of the forms: r2-i: Here "d" stands for the ramification index of the Puiseux's package.
The proof of Lemma 9.3 is ended. Let us complete the proof of Proposition 9.3. By Corollary 9.1 we know that Proposition 9.3 is true for a normalized coordinate change. By Remark 9.8 we know that it is true for a normalized Puiseux's package. We deduce the result for a general normalized transformation (A, ω) → (B, ω), by noting that we always have that ς B (ω) ≥ ς A (ω). 9.6. Resonances. Conditions "r1" and "r2" in Proposition 9.8 are the properties that may produce a stabilization of the critical height. Here we present them in terms of the reduced critical partω * and we describe their behaviour under normalized transformations.
• We say that (A, ω) satisfies the resonant property r1 if and only if d ≥ 2, χ = 1 andω * = ξx I zdΦ/Φ, with ξ ∈ k, ξ = 0 and I ∈ Z r ≥0 . and there are ξ = 0, I ∈ Z r , 0 = τ ∈ C r and υ ∈ C r such that one of the following expressions holds: Remark 9.10. Condition r2b-0 is equivalent tō In a general way, condition r2b-υ is equivalent tō Note that anyω * ∈ Ω 1 A written as in Equation (9.9) with I ∈ Z r satisfies automatically that I − χp ∈ Z r ≥0 , since the coefficient of dz is ξx I−χp (z − λx p ) χ−1 . We also have that r2b-υ is equivalent tō The reader can verify that conditions r1, respectively r2, coincide with the conditions r1, respectively r2, stated in Proposition 9.8 in terms ofᾱ.
Remark 9.11. Let us writeω * = χ s=0 z sω * s . Under the resonance conditions, the reduced critical levelω * χ is obtained as follows: There is an essential difference between the resonance conditions r1 and r2. In the case of r1, the ramification index is d ≥ 2 and in the case of r2, we have d = 1. On the other hand, the "bad resonance" r1 only occurs when χ = 1. In Propositions 9.9 and 9.10, we describe the possible transitions between the two types of resonance when χ = 1. Roughtly speaking, the resonance r1 occurs "at most once" during our local uniformization procedure.
Proof. Recall that the coordinate change is given by In this situation, we have thatω A 1 andω A ′ 1 have the same coefficients. More preciselȳ where A is a matrix of non-negative integer coefficients such that det A = 1, given by the property that ν(x µ ) = ν(x ⋆µA ).
In view of Equations (9.11), (9.12) and (9.13) we conclude that Statement (1) is true and it corresponds to the case ξ = 0.
As a consequence, if (A ⋆ , ω) satisfies the resonance condition r1, then we necessarily have that (A, ω) satisfies r2b-υ where υ is A-negative.
• It remains to prove Statement (4). We assume that (A, ω) satisfies r1. Hence the ramification index is d ≥ 2 and Φ = z d /x p . By Equation (9.11), we have The matrix C has not the form in Equation (9.14). Anyway, we have that This implies that ξ ⋆ = ξ ′ = 0 and τ ′ = 0, thus τ ⋆ = τ ′ A = 0. In particular we have neither r1 nor r2a. We get the resonance condition r2b-υ ⋆ , with υ ⋆ = p ⋆ and ν(z ′ ) = ν(x p ⋆ ) > 0. Thus υ ⋆ is not A ⋆ -negative and υ ⋆ = 0. 9.7. Never Ramified Cases and Horizontal Functions. Some of the results in this subsection may be considered as a truncated version of Classical Zariski's Local Uniformization [32] for the case of formal functions, although we add the property of respecting the normalized transformations used for the case of foliations. Definition 9.6. Consider (A, ω) with the property of χ-fixed critical height. We say that (A, ω) is never ramified if and only if for any normalized transformation (A, ω) → (B, ω) the (ℓ + 1)-ramification index of B is equal to one.
In next subsections we will reduce our study to never ramified cases. Let us define the horizontal coefficient H A,ω as follows. Write ω in A as Let H A,ω denote the coefficient h = ω(∂/∂z).
One important feature of never ramified situations is that the H A,ω is stable in the following sense: Lemma 9.9. Consider (A, ω) with the property of χ-fixed critical height and never ramified. We have: ) is a normalized Puiseux's package and Φ = z/x p is the contact rational function in A, then H B,ω = x p H A,ω .
Proof. The cases of a coordinate change or a γ-strict preparation are straighforward and left to the reader. If we consider a normalized Puiseux package, we necessarily have that resonant condition r2 holds, since we are non ramified and χ-fixed. The contact rational function is given by Φ = z/x p , where z B = Φ − λ. Noting that the result follows.
Let us consider a formal function F ∈ Ω 0 A . Looking at Definition 3.3, we recall A is a unit. We need a strong version of this concept, where we can assume thatF = 0. We precise it in Definition 9.7 below: A is a formal function in A. We say that (A, F ) is strongly γ-final if either ν A (F ) > γ or F = x I U , where U is a unit in Ω 0 A that can be written as Let us note that to be strongly γ-final is stable under any nested transformation.
Proof. We give quick indications of the proof, that follows the same lines of the case of 1-forms, without the difficulty of the preparation.
Let us note that the induction hypothesis allows us to make ρ-final any formal function G ∈ k[[x, y ≤ℓ ]] by means of an ℓ-nested transformation; to see this is enough to consider the differential dG, in view of Corollary 3.1. Moreover, these ℓ-nested transformations may be "integrated" in a γ-strict preparation of (A, ω), just by completing the γ-preparation.
We can perform a γ-preparation of F by means of an ℓ-nested transformation. To do this is enough to make γ-final one by one the levels F s with s ≤ γν(z). This can be done with a normalized transformation and hence we have ramification index equal to one. Now, we have a critical vertex at height ξ. Assume that the critical height does not drop. By a new normalized Puiseux's package, we get the property that the main vertex coincides with the critical vertex; moreover, this property remains true since we assume that the critical height does not drop. By making combinatorial independent blow-ups as in Proposition 5.3, we divide F by x I with ν(x I ) = ν A F and hence we obtain that F ξ is a unit. Up to performing additional ℓ-nested transformations, we can assume that for i = 0, 1. This property is stable under any new ℓ-nested transformation. This allows us to make a Tschirnhausen coordinate change By classical arguments, when we perform a new (A, ω)-normalized Puiseux's package A → B, either the critical height ξ drops ( when we are touching the "empty part" of the ξ − 1-level in the Newton Polygon ) or the value ν(z B ) ≥ min{γ, ν(z)}.
If ν(z B ) ≥ γ we are done, in the next transformation we get ς A (F ) ≥ γ. Otherwise, we see that ς B (F ) ≥ ς A (F ) + ν(z). We end in a finite number of steps.
Once F is γ-final, we write F = I x I F I . Now we perform a monomialization of the ideal generated by the x I by means of independent combinatorial blowups. This gives an expression of F as F = x I U , where U is a unit that we write U = ξ +Ũ andŨ is in the maximal ideal, by performing a normalized transformation containing a j-Puiseux's package for any j = 1, 2, . . . , ℓ + 1 we obtain that ν A (Ũ ) > 0, see Proposition 5.2. Proof. It follows from Lemma 9.9 and Proposition 9.11. Remark 9.13. Take (A, ω) with the property of χ-fixed critical height and never ramified. Assume that H A,ω is strongly γ-final and consider an (A, ω)-normalized transformation A → B. By Lemma 9.9, we see that We conclude that there is a normalized transformation (A, ω) → (B, ω) such that (B, ω) has γ-recessive or γ-dominant horizontal stability. 9.8. First Steps in the Reduction to Critical Height One. Let us start the proof of Proposition 9.4. We look for a contradiction with the existence of a γtruncated formal foliated space (A, ω) with the property of χ-fixed critical height, where χ ≥ 2. Thus, we assume we have such an (A, ω).
If (A, ω) → (B, ω) is a normalized transformation, then (B, ω) also have the property of the χ-fixed critical height. Then (B, ω) has the resonance property r2, since χ ≥ 2. In particular (A, ω) is never ramified, accordingly with Definition 9.6. By Corollary 9.2 and Remark 9.13, we can make the following assumption: A1: The horizontal coefficient H A,ω is strongly γ-final and (A, ω) has the γrecessive or γ-dominant horizontal stability. Note that A1 is stable under any new normalized transformations and thus (B, ω) also satisfies A1, when (A, ω) → (B, ω) is a normalized transformation.
Lemma 9.10. Let (A, ω) → (B, ω) be a normalized transformation that contains at least one normalized Puiseux's package. Then (9.15) where z B is the (ℓ + 1)-th dependent parameter in B. Moreover, the property in Equation (9.15) is stable under further normalized transformations.
Proof. We leave to the reader the details of this verification, based on the arguments in the proof of Proposition 9.3.
Remark 9.14. The property A2 is equivalent to saying that the critical vertex and the main vertex coincide.
Let us decompose ω into levels ω = s≥0 z s ω s , ω s = η s + h s dz/z, where we recall that h = s≥1 h s z s−1 .
Proof. This is obvious if ν A (h) > γ. When h = U x J , we have recalling that χ ≥ 2.
Lemma 9.12. The resonance condition r2a is satisfied for (A, ω). That is, the reduced critical partω * of ω has the form Proof. We already know that the resonant condition r2 is satisfied. We have only to show that condition r2b cannot occur. If r2b holds, the reduced partω * χ of the χ-level ω χ is given byω * see Equation (9.13). This implies that ν A (h χ ) = ν(x I−χp ) = ς A (ω) − χν(z). This is not compatible with the fact that ν A (h χ ) > ν(x I ) = ς A (ω) − χν(z) stated in Lemma 9.11. Then we have r2a and Equation (9.16) holds.
Lemma 9.14 below provides the key property we need to perform a "useful" Tschirnhausen transformation. Let us note that we need to invoke the γ-truncated integrability condition.
Proof. Let us consider the description of the truncated integrability condition given in Subsection 8.2. In particular, recall that ν A (∆ s ) ≥ 2γ, where .
In particular, there is J s such that J s = I i + I j when i + j = s. Then where we have ν A (ϑ s ) ≥ ǫ. Let us consider the following computation: Then we have ν A (Ξ s ) ≥ sν(z). Now, let us start the proof by finite induction on 1 ≤ s ≤ χ. If s = 1, we have Recalling that α 0 is 0-final dominant, we obtain U 1 andα 1 by Proposition 4.1.
In this subsection, we end the proof of Proposition 9.4. Let us take the notations and reductions in Subsection 9.8. We know that we may assume the following additional properties: A1: Then main vertex and the critical vertex coincide. This is equivalent to saying that ν A (ω) = ς A (ω) − χν(z). A2: H A,ω is strongly γ-final and (A, ω) has the γ-recessive or γ-dominant horizontal stability. A3: Up to performing a 0-nested transformation, for any 0 ≤ s ≤ χ, we have ω s = x Is β s , with β s being 0-final dominant and ν(x Is ) = ς A (ω) − sν(z). The above properties are stable under any further normalized transformations. Let us recall that a 0-nested transformation can be considered a normalized transformation, see Remark 9.2.
By Lemma 9.13, for 0 ≤ s ≤ χ we know that In particular, each α s is 0-final dominant. We also know that there are units Let us prepare the units U s to obtain a "Tschirnhausen coordinate change".
Let us perform an ℓ-nested transformation containing a j-Puiseux's package for each 1 ≤ j ≤ ℓ. Then each of the c IK x I y K ≤ℓ becomes a unit times a monomial in the independent variables, see Proposition 5.2. We principalize the list of of such monomials by using Proposition 5.3 and we are done.
We have just proved that (A, ω) does not satisfy the property of χ-fixed critical height, hence the proof of Proposition 9.4 is ended. 9.10. Critical Height One. Here we give a proof of Proposition 9.5. Thus, we assume that (A, ω) has the property of 1-fixed critical height and we look for a contradiction.
As a consequence of the study of the evolution of resonances, we can do a first reduction. Let us first consider the following definitions: • We say that (A, ω) is of an r2a-resonant persistent type if and only if (B, ω) is r2a for any normalized transformation (A, ω) → (B, ω). Proof. Let us recall the statements of Propositions 9.9 and 9.10. If we can find a normalized transformation (A, ω) → (B, ω) such that (B, ω) satisfies the resonant condition r2a, we get the persistent type r2a. If there is a normalized transformation (A, ω) → (B, ω) such that (B, ω) is r2b-υ, with υ being not B-negative, we obtain the persistent type r2b + . If we can get the condition r1, we are in the case r2b + after just one normalized Puiseux's package. The only remaining possibility is to be persistently in the case r2b-υ with "negative" υ, this is the case of the persistent type r2b × .
Let us make the following assumption from now on: P0: (A, ω) is of a persistent type r2a, r2b + or r2b × .
Remark 9.16. In particular, for any normalized transformation (A, ω) → (B, ω) the ramification index of B is equal to one. That is (A, ω) is never ramified.
With the same arguments as in Subsection 9.9, we assume the following additional properties (that are stable under any further normalized transformation): P1: Then main vertex and the critical vertex coincide. This is equivalent to saying that ν A (ω) = ς A (ω) − ν(z). P2: H A,ω is strongly γ-final and (A, ω) has the γ-recessive or γ-dominant horizontal stability. P3: Up to a 0-nested transformattion, for s = 0, 1, we have ω s = x Is β s , with β s being 0-final dominant and ν(x Is ) = ς A (ω) − sν(z). Once we have the above reductions, we are no more interested in performing normalized Puiseux's packages. We look for a contradiction with the existence of (A, ω) with the stated properties by performing only normalized coordinate changes. More precisely, we will contradict the property ς A (ω) < γ.
Recall that (A, ω) is r2a or r2b-υ, and that the contact rational function is Φ = z/x p . In view of Equation (9.10), we have that the following statements are equivalent From now on, we additionally assume the following property: P4: α 1 = α * 1 +α 1 , where dα * 1 = 0 and ν A (α 1 ) > 0. Proposition 9.12 below is the key observation we need to find the desired contradiction: Proposition 9.12. Assume that (A, ω) has the property of the 1-fixed critical height and properties P0, P1, P2, P3 and P4. There is a procedure to obtain a positive real number ǫ A (ω) > 0 from (A, ω) with the following property: there is a normalized coordinate change (A, ω) → (A ⋆ , ω) such that (A ⋆ , ω) satisfies the properties P0, P1, P2, P3, P4, and moreover Proof. In view of Equation (8.2), we have that ν A (∆ 1 ) ≥ 2γ, where ∆ 1 = η 1 ∧ η 0 + h 1 dη 0 + η 0 ∧ dh 1 . By expanding ∆ 1 , we obtain Let us consider the cases when (A, ω) is r2a, r2b + or r2b × . We know that they are independent situations and it is enough to show the existence of ǫ A (ω) with the desired properties in each of the three cases. Assume that (A, ω) is r2a. Let us see that ν A (H 1 ) ≥ 0. Since the reduced form ω * 1 of the level is given byω * 1 = dx τ /x τ , it is not possible for H 1 to be a unit, hence ν A (H 1 ) > 0, since h = H A,ω is strongly γ-final. Then, in this case α 1 is 0-final dominant. Let us fix ǫ 1 = ǫ A (ω) > 0 such that ǫ 1 ≤ min{ν(z), ν A (H 1 )}.
By applying finitely many times Proposition 9.12, we obtain a normalized transformation (A, ω) → (B, ω) such that ν B (z B ) > γ. But we know that This is a contradiction. In this way, the proof of Proposition 9.5 is ended.

Local Uniformization of Rational Foliations
In this section we show that Theorem 3 implies Theorem 1. More precisely, we show how to obtain pre-simple points as they were defined in the Introduction. In view of the results in [20], this is enough, since we can pass from pre-simple to simple points.
Before considering the proof of Theorem 1, we are going to develop a "nontruncated" version of the results in Section 9.
Proposition 10.1. Consider a stable (A, ω) with the property of non-truncated χ 0 -fixed critical height, where χ 0 ≥ 2. Up to performing a stable normalized transformation (A, ω) → (A ⋆ , ω) we get that (A ⋆ , ω) satisfies the following property: "For any stable normalized transformation (A ⋆ , ω) → (B, ω) and any γ ∈ R, we have that (B, H B (ω)) is not γ-final dominant." Proof. It is enough to work like in Subsections 9.8 and 9.9, with the following adaptations: • About the results in Subsection 9.8: We assumed that the horizontal coefficient H B (ω) was γ-final. Instead of that, we use that H B (ω) is a monomial times a unit. In Lemma 9.13, when we define the value ǫ, we must put ǫ = ν A (H B (ω)) − ς B (ω) + ν(z B ), instead of ǫ = min{γ, ν A (H B (ω))} − ς B (ω) + ν(z B ). • About results in Subsection 9.9: Every time ǫ appears, recall that it is defined without taking any value γ into account. In the last part of Subsection 9.9, we find a contradiction by increasing the value of the (l + 1)-th dependent parameter until ς B (n) (ω) > γ. Instead of this, the contradiction appears when ς B (n) (ω) − ν(z B (n) ) ≥ ν B (n) (H B (ω)).
Proposition 10.2. Consider a stable (A, ω) with the property of non-truncated 1-fixed critical height. There is a stable normalized transformation (A, ω) → (B, ω) such that (B, ω) has the resonance property r2a or r2b.
Remark 10.1. The cases of r2a or r2b with χ A (ω) = 1 correspond to pre-simple points, when the critical vertex coincide with the main vertex, up to perform independent blow-ups to allow a division by a monomial.

Rational
Foliations. Let us start with a projective variety variety M , with K = k(M ) and an rational codimension one foliation F ⊂ Ω K/k . Up to performing appropriate blow-ups, we can assume that the center P of R in M is a non-singular point of M and we have a locally parameterized model A = (O A ; x, y) adapted to R such that O M,P = O A . Now, we consider a nonzero differential 1-form ω ∈ Ω 1 A ∩ F . Let us remark that ω ∈ Ω 1 OA/k [log x] ⊂ Ω 1 A . Then ω is a Frobenius integrable rational differential 1-form. In particular, we see that the coefficients of ω belong to O A ⊂ K. Our objective is to perform blowups of M to get that ω satisfies the definition of pre-simple point given in the Introduction. The kind of blow-ups we perform are indicated by the corresponding allowed transformations of locally parameterized models, hence the centers of the blow-ups are non-singular and of codimension two, when we localise the situation at the center of the valuation.
We have a parameterized formal foliated space (A, ω), hence it is γ-truncated for any γ ∈ R. Thus we are in the conditions of Theorem 3, for any γ ∈ R. Applying Theorem 3, there are two possibilities: a) There are ρ ∈ R and an allowed transformation A → A ′ such that (A ′ , ω) is ρ-final dominant. b) For any ρ ∈ R, there is an allowed transformation A → A ′ such that (A ′ , ω) is ρ-final recessive.
In case a), we perform independent blow-ups A ′ → A ′′ , as in Proposition 5.3, to obtain that ω = x ′′ Iω , whereω ∈ Ω 1 A ′′ is 0-final dominant. This situation corresponds with a pre-simple corner as defined in the Introduction.
It remains to consider the case b). This case corresponds to a situation of "infinite value" of ω, in particular it does not appear when ω = df , for f ∈ O A . Nevertheless, it can happen for instance for an Euler's Equation, where the differential form is of polynomial type.
Let us work by induction on I A (ω), see Subsection 6.1. If I A (ω) = 0, we end by Corollary 5.2 and in fact we are in the case a). Assume I A (ω) = ℓ + 1, and take notations as in Subsection 6.1. Let us write where z is the last dependent parameter we consider. Recall that h ∈ O A ⊂ K and thus we have ν(h) < ∞, or h = 0. If h = 0, the integrability condition ω ∧ dω = 0 shows the existence of rational function F and a rational 1-differential form η such that I A (η) ≤ ℓ and ω = F η.
We end by induction.
Then, we can suppose that h = 0 and hence ν(h) < ∞. Consider the (formal) level decomposition For any index s, we can apply Theorem 4 to h s . We have two options: i) There are ρ ∈ R and an ℓ-nested transformation A → B such that (B, h s ) is ρ-final dominant. ii) For any ρ ∈ R, there is an ℓ-nested transformation A → B such that (B, h s ) is ρ-final recessive.
Then (A 1 , ω) is stable and we are able to apply Propositions 10.1 and 10.2. Up to a stable normalized transformation, we can assume that (A 1 , ω) has the property of χ 0 -fixed critical height.
The case χ 0 ≥ 2 does not happen. Indeed, by Proposition 10.1, the critical height cannot stabilize in χ 0 ≥ 2, since the horizontal coefficient h is nonzero and belongs to the local ring O A ⊂ K. Then, it has a well defined finite value and thus we can transform it into a unit times a monomial. To see this, we can apply Proposition 9.11 with respect to γ with γ > ν(h).
Note that χ 0 ≥ 1 since we are in case b), see also Proposition 9.2. If χ 0 = 1, by Proposition 10.2 and Remark 10.1 we get a pre-simple point.
The proof of Theorem 1 is ended.