Lipschitz Stratification of Complex Hypersurfaces in Codimension 2

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated, singularities in $\mathbb{C}^3$. We show that a natural stratification of such a family given by the singular set and the generic family of polar curves provides a Lipschitz stratification in the sense of Mostowski. In particular, such families are bi-Lipschitz trivial by trivializations obtained by integrating Lipschitz vector fields.


Introduction
In the geometric study of complex singular algebraic varieties or analytic spaces the notion of stratification plays an essential role. It is well known that there always exists a stratification that is topologically equisingular (i.e. trivial) along each stratum. This is usually achieved by means of a Whitney stratification. Another and entirely independent way of constructing such a stratification is Zariski equisingularity. A desirable important feature is the existence of a stratification that satisfies stronger equisingularity properties than the one given by Whitney Conditions. This is known about Zariski (generic) equisingularity, though its precise geometric properties are still to be understood. For instance, it is well known that Zariski equisingular families of plane curve singularities are bi-Lipschitz trivial. The goal of this paper is to extend this observation to the next case, the families of surface singularities in C 3 .
In 1979 O. Zariski [29] presented a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. Zariski's theory is based on the notion of equisingularity along the strata defined by considering the discriminants loci of successive "generic" projections. This concept, now referred to as Zariski equisingularity or generic Zariski equisingularity, was called by Zariski himself algebro-geometric equisingularity, since it is defined by purely algebraic means but reflects several natural geometric properties. In [27] Zariski studied the case of strata of codimension one. In this case the hypersurface is locally isomorphic to an equisingular (topologically trivial if the ground field is C) family of plane curve singularities. Moreover, by Theorem 8.1 of [27], Zariski's stratification satisfies Whitney's conditions along the strata of codimension one, and over C, by [18], such an equisingular family of plane curves is bi-Lipschitz trivial, i.e. trivial by a local ambient bi-Lipschitz homeomorphism. In general, Zariski equisingularity implies Whitney's conditions as shown by Speder [20]. For a survey on Zariski equsingularity and its recent applications see [16].
In 1985 T. Mostowski [9] introduced the notion of Lipschitz stratification of complex analytic spaces or algebraic varieties, by imposing local conditions on tangent spaces to the strata, stronger than Whitney's conditions. Mostowski's work was partly motivated by the question of Siebenmann and Sullivan [19] whether the number of local Lipschitz types on (real or complex) analytic spaces is countable. Mostowski's Lipschitz stratification satisfies the extension property of stratified vector fields from lower dimensional to higher dimensional strata, and therefore implies local bi-Lipschitz triviality. Its construction is similar to the one of Zariski, but involves considering many projections at each stage of construction. It is related to the geometry of polar varieties, as shown by Mostowski in the case of hypersurface singularities in C 3 , see [10]. In general, the construction of a Lipschitz stratification is complicated and involves many stages. It was conjectured by J.-P. Henry and T. Mostowski that Zariski equisingular families of surface singularities in C 3 admit natural Lipschitz stratification by taking the singular locus and the family of "generic" polar curves as strata. We show this conjecture in this paper, see Theorem 2.1.
Recent works, see for instance [23], [12], [5], [6] [3], show further development and progress on understanding the Lipschitz structure of singularities and its relation to other geometric phenomena appearing in the study of local properties of complex or real analytic or algebraic singular spaces. Among the major results and contributions we mention only the most important ones related to this paper, [1] where the case of the "inner" metric was considered and [11] where the equivalence of Zariski Equisingularity and Lipschitz triviality for families of complex normal surface singularities was announced.
Our proof of Theorem 2.1 is based on local parameterizations of two geometric objects associated to such families: the polar wedges and the quasi-wings. Both originate from the classical wings introduced by Whitney in [25]. The polar wedges are neighborhoods of families of polar curves, the critical loci of corank-one projections. The quasi-wings, originally introduced in [9], are neighbourhoods of curves on which this projection is regular (with a control on the derivatives). Their local parameterizations, interesting by themselves, in the case of polar wedges originate from [2] and [22] and were recently considered in [11].
As we show the quasi-wings and the polar wedges cover a neighbourhood of the singularity. The proof of this fact follows from the analytic wings construction of [17].
The definition of "generic projection" is crucial for Zariski's theory. Zariski's study of codimension one singularities (families of plane curve singularities) required just transverse projections. This is no longer the case for singularities in codimension 2. In [7] Luengo gave an example of a family of surface singularities in C 3 that is Zariski equisingular for one transverse projection but not for a generic transverse projection. Therefore we make precise what we mean by "generic projection" in our context and we state it in our Transversality Assumptions. This is important since this condition can be computed and algorithmically verified.
2.1. Zariski equisingularity. Given a family of reduced analytic functions germs f t (x, y, z) : (C 3 , 0) → (C, 0) as above, we denote by ∆(x, y, t) the discriminant of the projection π restricted to X . The zero set of ∆(x, y, t) is a family of plane curve singularities parameterized by t. We say that the family X t is Zariski equisingular (with respect to the projection π) if t → {∆(x, y, t) = 0} is an equisingular family of plane curves, that is satisfying one of the standard equivalent definitions, see [26], [21, p. 623]. We shall often use the classical result saying that a family of equisingular plane curves admits a uniform Puiseux expansion with respect to parameters, in the sense of [17,Theorem 2.2]. We refer to it as to the Puiseux with parameter theorem.
We say that the family X t is generically linearly Zariski equisingular if it is Zariski equisingular after a generic linear change of coordinates x, y, z.
In the proof of Theorem 2.1 we use the following precise assumptions on f , called Transversality Assumptions, that are implied by the generic linear Zariski equisingularity.
Let us denote by π b the projection C 3 × C l → C 2 × C l parallel to (0, b, 1, 0), that is π b (x, y, z, t) = (x, y − bz, t). We denote by ∆ b (x, y, t) the discriminant of the projection π b restricted to X .
Transversality Assumptions. The tangent cone C 0 (X 0 ) to X 0 = f −1 0 (0) does not contain the z-axis and, for b and t small, the family of the discriminant loci ∆ b = 0 is an equisingular family of plane curve singularities with respect to b and t as parameters. Moreover, we suppose that ∆ 0 = 0 is transverse to the y-axis and that x = 0 is not a limit of tangent spaces to X reg , the regular part of X .
Remark 2.2. Since Zariski equisingular families are equimultiple, see [28] or [17] [Proposition 1.13], the above assumptions imply the following. The tangent cone C 0 (X t ) does not contain (0, b, 1), for t and b small. The y-axis is transverse to every {(x, y); ∆ b (x, y, t) = 0}, also for t and b small.
We now show that a generically linearly Zariski equisingular family satisfies, after a linear change of coordinates x, y, z, the Transversality Assumptions. First we need the following lemma. Lemma 2.3. The family f t (x, y, z) = 0 is generically linearly Zariski equisingular if and only if, after a linear change of coordinates x, y, z, the family f (x + az, y + bz, z, t) = 0, for a, b, t small, is Zariski equisingular with respect to parameters a, b, t.
Proof. The "if" part is obvious. We show the "only if". Let ∆(x, y, a, b, t) be the discriminant of f (x + az, y + bz, z, t). By assumption there is an open subset U ⊂ C 2 such that this family of plane curve germs ∆(x, y, a, b, t) = 0 is equisingular with respect to t for every (a, b) ∈ U. Fix a small neighbourhood V of the origin in C l so that the subset of parameters (a, b, t) ∈ U × V, such that ∆(x, y, a, b, t) = 0 changes the equisingularity type, is a proper analytic subset of Y ⊂ U × V . The existence of such Y follows for instance from Zariski [26], where it is shown that a family of plane curve singularities is equisingular if and only if its discriminant by a transverse projection is equimultiple.(Equivalently, one may use semicontinuous invariants characterizing equisingularity such as the Milnor number for instance.) Then Y cannot contain U × {0} (this would contradict the Zariski equisingularity of ∆ = 0 for (a, b) ∈ U arbitrary and fixed). Therefore, the family f (x + az, y + 5 bz, z, t) = 0 is Zariski equisingular for the parameters a, b, t in a neighborhood of any point of (U \ Y ) × {0}. This shows the claim.
Suppose now that the family f t = 0 is generically linearly Zariski equisingular and choose a generic line ℓ in the parameter space of (a, b) ∈ U in the notation of the proof of the above lemma. The pencil of kernels of π a,b (x, y, z, t) = (x − at, y − bz, t), (a, b) ∈ ℓ, corresponds to a hyperplane H ⊂ C 3 . Choose coordinates x, y, z so that H = {x = 0} and then ℓ corresponds to the pencil of projections parallel to (0, b, 1) ∈ H. Then in this system of coordinates (x, y, z), f satisfies the Transversality Assumptions.
2.2. Lipschitz stratification. In [9] T. Mostowski introduced a sequence of conditions on the tangent spaces to the strata of a stratified subset of C n that, if satisfied, imply the Lipschitz triviality of the stratification along each stratum. Mostowski showed the existence of such stratifications for germs of complex analytic subsets of C n . Note that there is no canonical Lipschitz stratification in the sense of Mostowski in general.
In [10] Mostowski gave a criterion for a set to be a codimension one stratum of Lipschitz stratification of a complex surface germ in C 3 , see the second example on pages 320-321 of [10]. This criterion implies that a generic polar curve can be chosen as such a stratum. It is not difficult to complete Mostowski's argument and show Theorem 2.1 in the non-parameterized case (l = 0). In subsection 6.1 we give a different proof which implies the parameterized case as well.
Mostowski's conditions imply the existence of extensions of Lipschitz stratified vector fields from lower dimensional to higher dimensional strata, the property which, as shown in [13], is equivalent to Mostowski's conditions. Let us recall this equivalent definition. For this it is convenient to express Mostowski's stratification in terms of its skeleton, that is the union of strata of dimension ≤ k. Let X ⊂ C n be a complex analytic subset of dimension d and let l ≥ 0, X l−1 = ∅, be its filtration by complex analytic sets such that every X k \ X k−1 is either empty or nonsingular of pure dimension k.
Our proof is based on the following characterization of Lipschitz stratification. (i) There exists C > 0 such that for every W ⊂ X satisfying X j−1 ⊆ W ⊂ X j , every Lipschitz stratified vector field on W with a Lipschitz constant L, bounded on W ∩ X l by K, can be extended to a Lipschitz stratified vector field on X j with a Lipschitz constant C(L + K). (ii) There exists C > 0 such that for every W = X j−1 ∪ {q}, q ∈ X j , each Lipschitz stratified vector field on W with a Lipschitz constant L, bounded on W ∩ X l by K, can be extended to a Lipschitz stratified vector field on W ∪ {q ′ }, q ′ ∈ X j , with a Lipschitz constant C(L + K).
Here by a stratified vector field we mean a vector field tangent to strata. In our particular case, stratification {X \ S, S \ T, T } it Lipschitz if and only if there is a constant C > 0 such that: (L1) for every couple of points q, q ′ ∈ S \ T , every stratified Lipschitz vector field on T ∪ {q}, with Lipschitz constant L and bounded by K, can be extended to a Lipschitz stratified vector field on T ∪ {q, q ′ } with Lipschitz constant C(L + K). (L2) for every couple of points q, q ′ ∈ X \ S, every stratified Lipschitz vector field on S ∪ {q} with Lipschitz constant L and bounded by K, can be extended to a Lipschitz vector field on S ∪ {q, q ′ } with Lipschitz constant C(L + K). In order to show the conditions (L1) and (L2) we consider two geometric constructions, the quasi-wings of Mostowski [9] and the polar wedges of [1] and [11], that, as sets, together cover the whole X . We first show the (L1) condition in general and the (L2) condition on polar wedges. This part of the proof is based on a complete description of the stratified Lipschitz vector fields on polar wedges in terms of their parameterizations, see Section 5. Note that in order to compare points on polar wedges we work with fractional powers, using parameterizations over the same allowable sector, see the Subsection 4.1 for more details. In order to show (L2) on the quasi-wings we employ the following strategy. If Mostowski's conditions fail then they fail along real analytic arcs γ(s), γ ′ (s), s ∈ [0, ε), see [9] Lemma 6.2 or the valuative Mostowski's conditions of [6].
For such arcs, however, if they are not in the union of polar wedges, we can construct quasi-wings containing them, say QW and QW ′ respectively, and then we show that the stratification {QW ∪ QW ′ \ S, S \ T, T } satisfies criterion (L2) on the arcs γ(s), γ ′ (s). For a precise statement and proof justifying this strategy the reader is referred to the rather technical Section 11.

Notation and conventions.
In what follows we often use the following notations. For two complex function germs f, g : (C k , 0) → (C, 0) we write : |g(x)| → 0 as x → 0. While parameterizing analytic curve singularities or families of such singularities in C 2 and C 3 using Puiseux Theorem, we ramify in variable x = u n . We often have to replace such an exponent n by a multiple in order for such parameterizations to remain analytic, but we keep denoting it by n for simplicity. This makes no harm since we always work over an admissible sector as explained in subsection 4.1. By an analytic unit we mean a nowhere vanishing analytic function or a germ of such function.

Families of polar curves
In this section we discuss how the families of polar curves of X , associated to the projections π b , b ∈ C, depend to b. The main result is Proposition 3.3 (nonparameterized case) and Proposition 3.4 (parameterized case). The proposition in the non-parameterized case appeared in the proof of the Polar wedge lemma, i.e. Proposition 3.4, of [1]. The proofs of Propositions 3.3 and 3.4 are based on a key Lemma 3.1 due to [2] and [22].
3.1. Non-parameterized case. For simplicity we first consider the case of f (x, y, z) without parameter. We assume that the coordinate system satisfies the Transversality Assumptions and therefore the family parameterized by b ∈ C is Zariski equisingular for b small. By this assumption the zero set of the discriminant ∆ F (X, Y, b) of F satisfies the Puiseux with parameter theorem. The set F = F ′ Z = 0, is the union S F = Σ F ∪ C F of the singular set Σ F of F and the family of the polar curves C F . The set S F consists of finitely many irreducible components parameterized by The below key lemma is a version of the first formula on page 278 of [2] or of a formula on page 465 of [22].
We differentiate the first identity with respect to b and use the second one to simplify the result If f ′ y (u n , Y i + bZ i , Z i ) ≡ 0 then the formula (5) holds. Note that in this case (4) parameterizes an irreducible component of C F . If f ′ y (u n , Y i + bZ i , Z i ) ≡ 0 then, in addition to (6), we have F ′ Y (u n , Y i , Z i , b) = 0. Thus in this case (4) parameterizes a component of Σ F . By the formula (X, Y, Z, b) ∈ Σ F if and only if (x, y, z) = (X, Y + bZ, Z) ∈ Σ f , the singular set of f . Thus in this case the map parameterizes a component of Σ f . Moreover, by the Transversality Assumptions, the projection of Σ f on the x-axis is finite. Consequently, both y i = Y i + bZ i , and Z i are independent of b and (5) trivially holds.
We note that, if f ′ y (u n , Y i + bZ i , Z i ) ≡ 0, i.e. if (4) parameterizes a component of C F , then (8) parameterizes a family of polar curves in f −1 (0) defined by the projections π b . In both cases, the functions y i (u, b), z i (u, b) = Z i (u, b), and Y i (u, b) are related by 8 In particular, the expansion of y i cannot have a term linear in b.
By the Zariski equisingularity assumption for any two distinct branches (u, b). Note that, by the transversality with the y−axis, we have k ij ≥ n . By (9) this implies the following result.
The next result, that we will prove later in the more general parameterized case, is crucial.
3.2. Parameterized case. We extend the results of the previous subsection to the parameterized case family with f satisfying the Transversality Assumptions. Thus F is now Zariski equisingular with respect to the parameters b and t and therefore the discriminant ∆ f (X, Y, b, t) of F with respect to Z satisfies the Puiseux with parameter theorem. Similarly to the non-parameterized case, and consists of the singular locus Σ F and a family C F of polar curves, now parameterized by b and t.
The lemma 3.1 still holds (with the same proof) so we have Z i = −∂Y i /∂b. Then parameterize in C 3 × C l the families of polar curves with respect to the projections π b with t being a parameter, or the branches of the singular locus Σ f . The relations (9) are still satisfied.
Also the counterpart of Proposition 3.3 holds. We give its proof below.
Proof. If y i (u, b, t) and z i (u, b, t) are independent of b then (14) parameterizes a branch of the singular locus of Σ f . Therefore we suppose that one of them, and hence by (9) both of them, depend notrivially on b. Expand ∂z i ∂b (u, b, t) = k≥m a k (b, t)u k with a m (b, t) ≡ 0. To show the result it suffices to show that a m (0, 0) = 0.
Proof. Using (15) we get Differentiating with respect to b and applying (9), we conclude that (ϕ i (u, b, t) − ψ i (u, b, t)) is a unit (as ψ i is unit by (15)).
The following lemma follows from the Zariski equisingularity assumption. Lemma 3.6.

Polar wedges
In this section we consider the polar wedges in the sense of [1] and [11]. The polar wedges are neighbourhoods of the polar curves that play a crucial role in our proof of Theorem 2.1. The formal definition is the following. Definition 4.1 (Polar wedge). We call a polar wedge and denote it by PW i the image of the map p i (u, b, t) defined by (14) (for |b| < ε with ε > 0 small), that parameterizes a family of polar curves associated to the projections π b . (14) is independent of b, that is it parameterizes a branch of the singular set Σ f , then it does not define a polar wedge. Two polar wedges (defined for the same ε) either coincide as sets or are disjoint for u = 0. Moreover, 4.1. Allowable sectors. Let PW i be a polar wedge parameterized by p i and let θ be an n-th root of unity. Then p i (θu, b, t) could be identical to p i (u, b, t) or not, but it always parameterizes the same polar wedge as a set. In order to avoid confusion and also to compare two different polar wedges we work over allowable sectors. We say that a sector Ξ = Ξ I = {u ∈ C; arg u ∈ I} is allowable if the interval I ⊂ R is of length strictly smaller than 2π/n. If we consider only u ∈ Ξ then x = u n = 0 uniquely defines u. That means that over such an x, every point in the union of polar wedges is attained by a unique parameterization.
Therefore we may write such parameterization (14) in terms of x, b, t assuming implicitly that we work over a sector Ξ Remark 4.2. We note that any two points in polar wedges p i (u 1 , b 1 , t 1 ) and p j (u 2 , b 2 , t 2 ) can be compared using parameterizations over the same allowable sector. Indeed, given nonzero u 1 , u 2 there always exists an n-th root of unity θ and an allowable sector Ξ that contains u 1 and θu 2 and an index k such that p j (u 2 , b 2 , t 2 ) = p k (θu 2 , b 2 , t 2 ).

Distance in polar wedges.
Having an allowable sector fixed we show below formulas for the distance between points inside one polar wedge and the distance between points of different polar wedges. Note that these formulas imply, in particular, that different polar wedges do not intersect outside T = {x = y = z = 0}. In order to avoid heavy notation we do not use special symbols for the restriction of a polar wedge to an allowable sector.

For every pair of polar wedges
Proof of Proposition 4.3. We divide the proof in four steps. In the first two steps we reduce the proofs of all (21), (22), (23) to simpler cases. In particular, while considering the formula (21) we suppose below that i = j.

First reduction.
We claim that it suffices to prove the formulas (21), (22), (23) for t 1 = t 2 . This follows from that we show now. The first property is obvious, . This completes the proof of first reduction claim.

Second reduction.
We claim that it suffices to show the formulas of the above proposition for the case t = t 1 = t 2 , x 1 = x 2 . The argument is similar to the one above.The property Lemma 4.6. We have for each i and similar bounds hold for z i in place of y i .
Proof. If (u 1 , b, t), (u 2 , b, t) are in the same allowable sector then we have 12 that is both sides are comparable up to a constant depending only on the sector. Denote y i (u, b, t) = u nŷ i (u, b, t) and suppose |u 2 | ≥ |u 1 |. Then This shows the first formula; the second one can be shown in a similar way. (21) and (22). We assume t = t 1 = t 2 , x = x 1 = x 2 . Then (21) follows from (15) and the fact that b → bψ(b) is bi-Lipschitz (ψ a unit), and (22) follows from

Proof of
Now (23) follows from (24), (25). Indeed, we may consider separately the three cases: |x| k i,j /n ∼ |b 1 − b 2 ||x| m/n , |x| k i,j /n dominant, and |b 1 − b 2 ||x| m/n dominant, and suppose that b 1 , b 2 are small in comparison to the units.

Stratified Lipschitz vector fields on polar wedges
In this section we describe completely the stratified Lipschitz vector fields on polar wedges in terms of their parameterizations. Note that these descriptions are valid only over allowable sectors, see Remark 4.2.
Let PW i be a polar wedge parameterized by (14). We call the polar set C i , parameterized by p i (u, t) := p i (u, 0, t), the spine of PW i . A vector field on PW i is stratified if it is tangent to the strata: T , C i \ T , and to PW i \ C i .

Stratified Lipschitz vector fields on a single polar wedge. Let
We always suppose the vector field p i * (v) is well defined on PW i , that is independent of b if x = 0, and it is stratified, that is tangent to T and C i \ T ; The independence on b if x = 0 implies that both α(0, b, t) and β(0, b, t) are independent on b, and the actual tangency to T assures that in fact β(0, b, t) = 0. The tangency to C i \ T implies δ(x, 0, t) = 0. We also note that p i * ( ∂ ∂b ) is always zero on x = 0. 13

Suppose that a function
Proof. We show that each coordinate of these vector fields is Lipschitz. For this computation it is more convenient to use the parameter u instead of x since these vector fields are analytic in u, b, t. For clarity we also drop the index i coming from the parameterization (14).
The t-coordinate of p * ( ∂ ∂t ) equals 1 = ∂t ∂t and is Lipschitz. The x-coordinate of p * ( ∂ ∂t ) vanishes identically. Let us show, using Proposition 3.4 and Lemma 4.6, that the y-coordinate of p * ( ∂ ∂t ) is Lipschitz (the argument for the z coordinate is similar) All the other cases can be checked in a similar way.
Proof. If p i * (v) is Lipschitz then so is its t-coordinate, that is α. We claim that if α satisfies (26) so do α ∂y i ∂t and α ∂z i ∂t . This follows from Proposition 5.1 because the product of two Lipschitz functions is Lipschitz. This shows that p i * (α ∂ ∂t ) is Lipschitz. By subtracting it from p i * (v) we may assume that α ≡ 0. If . Then, by (21) in Proposition 4.3 and the Lipschitz property between p i (x, b, t) and p i (0, b, t), we have |β| |x| as claimed.
To use a similar argument to the previous "the product of Lipschitz functions is Lipschitz", we need the following elementary generalization. Lemma 5.3. Suppose h : X → C is a Lipschitz function on a metric space X and let L h := {f : X → C; Lipschitz on X , |f | |h|}. If f, g ∈ L h , then ξ := f g/h ∈ L h (here ξ is understood to be equal to 0 on the zero set of h).

5.2.
Lipschitz vector fields on the union of two polar wedges. Consider two polar wedges PW i and PW j parameterized by p i (x, b, t) and p j (x, b, t), over the same allowable sector, see 4.1 for more details.
Leth be a function defined on a subset of PW i ∪PW j by two functions h k (x, b, t), k = i, j. Then, after Proposition 4.3,h is Lipschitz iff so are its restrictionsh i and h j to PW i and PW j respectively, and where m = min{m i , m j }.
Proposition 5.4. The vector fields given by , with q analytic, see Lemma 3.6.
For k = i, j let p k * (v k ) be a vector field on a subset of W Ξ,k given by Proposition 5.5. The vector field given by p k * (v k ), k = i, j, defined on a subset U of PW i ∪ PW j containing C i ∪ C j , is stratified Lipschitz iff the following conditions are satisfied: 0) each p k * (v k ) is stratified Lipschitz on U ∩ PW k ; 1) α i , α j satisfy (27); 2) β i , β j satisfy (27); 3) δ i x m/n , δ j x m/n satisfy (27).
Proof. The proof is similar to the proof of Proposition 5.2 and it is based on Lemma 5.3 and Proposition 5.4.
Remark 5.6. Ifh i ,h j are stratified Lipschitz on PW i and PW j respectively, then, by Corollary 4.4, it suffices to check (27) for t = t 1 = t 2 , u = u 1 = u 2 , and b = b 1 = b 2 . Therefore, in this case, (27) can be replaced by Thus, for j = i, the functions β j and δ j are defined only for b = 0 (and hence δ j = 0 since the vector field is stratified). The functions β i and δ i are defined on . By Propositions 5.2 and 5.5 it suffices to extend β j and δ j to two families of functions, still denoted by β j , δ j , that satisfy the conditions given in those propositions. For all j we define Then, because |β 0 − β i (x 0 , 0)| ≤ CL|b 0 ||x 0 | m i /n , where L is the Lipschitz constant of the vector filed v and C is a universal constant, the first summand of the right-hand side of (29) satisfies 2) of Propositions 5.2 and 5.5. The argument for (30) is similar because |δ 0 | ≤ CL|b 0 |. This completes the proof of Theorem 2.1 for PW in the non-parameterized case.
6.2. Parameterized case. By Corollary 4.5 and Propositions 5.2, 5.5, the map given X 0 × T → X , restricted to PW ∩ X 0 , defined in terms of the parameterizations of polar wedges by (p i (0, x, b), t) → p i (x, b, t), is not only Lipschitz but also establishes a bijection between the Lipschitz vector fields. Therefore, by Proposition 2.4, {PW \ S, S \ T, T } is a Lipschitz stratification if and only if so is its intersection with X 0 and the latter is a Lipschitz stratification by the non-parameterized case. We use here an easy observation that the cartesian product of a Lipschitz stratification by a smooth space is also Lipschitz (actually the cartesian product of two Lipschitz stratifications is Lipschitz).
6.3. Examples. In [10] Mostowski gives a criterion for the codimension 1 stratum of Lipschitz stratification. In particular he proposes the following example (we change the order of variables so it follows our notation): f (x, y, z) = z 2 − (y 3 + y 2 x 2 ). The singular set Σ f of X = {f = 0} is the x-axis but as Mostowski shows is not a Lipschitz stratification of X. By solving the system f = ∂f /∂z − b∂f /∂y = 0 one can check that there is one polar wedge with n = 1 and m = 4 given by and one has to add a generic polar curve, or just a curve y = −x 2 + b 2 x 4 + · · · , z = 3bx 4 + · · · , to Σ f to get the one dimensional stratum. In [10, Section 7] Mostowski studies the case of surface singularities in C 3 and shows in particular the following result. We give below an alternative proof of this proposition.
Proof. Let q 0 = p(x 0 , b 0 ) ∈ X \{0} be on the polar wedge parameterized by p(x, b) = (x, y(x, b), z(x, b)), x = u n , where y, z are as in (11). Let v 0 = p * ( ∂ ∂b ) be the vector tangent at q 0 = p(x 0 , b 0 ) to X. We extend it by 0 to {0} and get a Lipschitz vector field on {0} ∪ {q 0 } with Lipschitz constant L = Cx m/n−1 0 , where C > 0 depends only on the polar wedge. Suppose we extend this vector field to q 1 = p(x 1 , b 1 )), ∂ ∂b ) so that the extended vector field has Lipschitz constant L 1 = C 1 L. By the Lipschitz property of the x-coordinate of this vector field |α 1 | ≤ C 1 L q 0 − q 1 ∼ C 1 L|b 0 − b 1 ||x 0 | m/n . Therefore, we can subtract from v 1 the vector p * (α 1 ∂ ∂x ) without changing significantly the Lipschitz constant (just changing C 1 ). Thus we may assume that α 1 = 0. By the Lipschitz property of the y and z-coordinates of this vector field whereφ,ψ are units. Considering (31) as a system of linear equations with the unknowns 1 (in front of the first summands of both equations) and δ 1 , by Cramer's rule, |, that is impossible if we allow x 0 → 0, as by our assumption m > n.

Quasi-wings
Quasi-wings were introduced by Mostowski in [9,Section 5] in order to show the existence of Lipschitz stratification in complex analytic case. In this construction Mostowski used several co-rank one projections, instead of a single one, to cover the whole complement of Σ f in X by quasi-wings. We use the quasi-wings to study Lipschitz vector fields on the complement of PW.
The main idea of construction goes as follows (the details will follow later). Given a real analytic arc p(s), s ∈ [0, ε), of the form p(s) = (s n , y(s), z(s), t(s)), Our goal is to embed p(s) in a quasi-wing QW (kind of cuspidal neighborhood of p(s) in X ), that is the graph of a root of f over a set W q , the image of where u, v ∈ C are supposed small. Geometrically, W q is a cuspidal neighborhood of π(p(s)), that we call a wedge, and QW is its lift to X . Thus QW admits a parameterization of the form p(u, v, t) = (q(u, v, t), z(u, v, t)) such that p(s) = p(s, 0, t(s)). We shall make the following assumptions on p(s) : (1) p(s) is not included in S and moreover for every polar branch C i there is an exponent l i such that s l i ∼ dist(p(s), C i ) ∼ dist(π(p(s)), π(C i )). A similar assumption is made on every branch of the singular locus Σ f . In particular we have dist(p(s), S) ∼ dist(π(p(s)), π(S)). (3) sl dist(p(s), S) ∼ dist(π(p(s)), π(S)), that is QW does not touch S (except along T ), and this property is preserved by the projection to the t, x, y-space. Then PW ∩ QW is just the T stratum and as we show in Proposition 7.3 (4) QW is the graph of a root of f whose all first order partial derivatives are bounded. In particular, the projection QW → W q is bi-Lipschitz.
In the formal definition of quasi-wings we will require thatl is chosen minimal for (3), i.e. sl ∼ dist(p(s), S) ∼ dist(π(p(s)), π(S)), (we seek the maximal possible set W q satisfying the above properties). We show in Proposition 7.7 that each real analytic arc satisfying (1) and (2) can be embedded in a quasi-wing. In general, any real analytic arc that is not embedded in the singular locus, satisfies the conditions (1) or (2) after a small linear change of coordinates and therefore can be embedded in a quasi-wing in this new system of coordinates, see Corollary 7.8. We note that our construction of quasi-wings differs significantly from the one of Mostowski. We use the Puiseux with parameter theorem and arc-wise analytic trivializations of [17]. The latter also provides a crucial partial Lipschitz property, see Remark 7.6 that we use in the proof of Proposition 7.7. Consequently, our construction of quasi-wings can be extended to the real analytic set-up. Mostowski uses instead the bound on derivatives of holomorphic functions (Schwarz's Lemma).
7.1. Regular wedges and quasi-wings. Let ∆(x, y, t) denote the discriminant of f (x, y, z, t). The discriminant locus ∆ = 0 is the finite union of families of analytic plane curves parameterized by (u, t) → (u n , y i (u, t), t).
By the Zariski equisingularity assumption we have and by the Transversality Assumptions y i (u, t) = O(u n ). Note that y i of (33) is either the projection of a polar branch, the one denoted by y i (u, 0, t) in (15) and from now on it will be indexed by i ∈ I C , or parameterizes the projection of a branch of the singular locus Σ f , and it will be indexed by i ∈ I Σ .
We assume y(u, t) = O(u n ) and that for each discriminant branch (33), y(u, t) satisfies, for some integersl i , y(u, t) − y i (u, t) = ul i unit(u, t). Remark 7.1. As both y(u, t) = O(u n ) and y i (u, t) = O(u n ) it follows thatl i ≥ n.
Consider the map q(u, v, t) = (u n , y(u, t) + ulv, t), defined for complex v, |v| < ε with ε > 0 small, and denote its image by W q . We supposel ≥ max ili , that is the image of q, for u = 0, is inside the complement of the discriminant locus ∆ = 0. Lemma 7.2. Let g(u, v, z, t) = f (q(u, v, t), z). Ifl ≥ max ili then the discriminant of g satisfies Then, by assumptionl ≥ max ili , Therefore, by Puiseux with parameter theorem, after a ramification in u, we may assume that the roots of g are analytic functions of the form z τ (u, v, t) = z τ (u n , y(u, t) + vul, t) and that for every pair of such roots Moreover, by transversality of projection π, z τ (u, v, t) = O(u n ). Proposition 7.3. Supposel i ≤ m i for every projection (33) of a polar branch. Then the (first order) partial derivatives of the roots z τ (x, y, t) of f over W q (the image of (35)), are bounded. Therefore, the roots of g are of the form Finally, ∂ ∂y (z τ (x, y, t)) is bounded on W q by the conditionsl i ≤ m i ,l i ≤l, and (15). Indeed, since f (x, y, z τ (x, y, t), t) ≡ 0 we have on the graph of z τ If | ∂zτ ∂y | > N, then, by (7), the graph of z τ (x, y, t) on W q would intersect a polar wedge PW i for b = ( ∂zτ ∂y ) −1 . This is only possible ifl i ≥ min{l, m i }. Ifl i = min{l, m i } then this intersection is empty provided we suppose both b and v sufficiently small (and hence N large).
We introduce now a version of quasi-wings and nicely-situated quasi-wings of [9]. Definition 7.4 (Quasi-wings). We say that the image of q(u, v, t) of (35) is a regular wedge W q ifl = max i∈I C ∪I Σl i and ifl i ≤ m i for every i ∈ I C . Then by a quasi-wing QW τ over W q we mean the image of an analytic map p τ (u, v, t) = (q(u, v, t), z τ (u, v, t)), where z τ is a root of f (q t (u, v), z).
We say that two quasi-wings QW τ , QW ν are nicely-situated if they lie over the same regular wedge W q . 7.2. Construction of quasi-wings. Consider a real analytic arc p(s), s ∈ [0, ε), of the form p(s) = (s n , y(s), z(s), t(s)), π(p(s)) = q(s) = (s n , y(s), t(s)), Under some additional assumptions we construct in Proposition 7.7 a quasi-wing containg the arc p(s). For this we use in the proof of Lemma 7.5 the arc-wise analytic trivializations of [17] and construct, following [17, Proposition 7.3], of a complex analytic wing containing q(s).
Let (u n , y i (u, t), z i (u, t), t), i ∈ I C , be a parameterization of the polar branch C i , and let (u n , y k (u, t), z k (u, t), t), k ∈ I Σ , be a parameterization of the branch Σ k of the singular set Σ f . 20 Lemma 7.5. Let q(s) = (s n , y(s), t(s)), y(s) = O(s n ), be a real analytic arc at the origin. For each polar branch C i , parameterized as above, denote q i (u, t) = (u n , y i (u, t), t) and letl i = ord s (y(s) − y i (s, t(s)). Then there is a complex analytic wing parameterized by q(u, t) = (u n , y(u, t), t), y(u, t) = O(u n ) containing q(s), that is satisfying y(s) = y(s, t(s)), such that y(u, t) − y i (u, t) equals ul i times a unit. In particular, over the same allowable sector we have and ord s dist(q(s), π(C i )) =l i .
Proof. By [17,Theorem 3.3] there is an arc-wise analytic local trivialization Φ : is complex analytic with respect to t, and both Φ and its inverse Φ −1 are real analytic on real analytic arcs. By [17, Proposition 3.7] we may require Ψ 1 (x, t) = x, so the allowable sectors are preserved.
By the arc-analyticity of Φ −1 , there exists a real analytic arc (s n ,ỹ(s), t(s)) such that Φ(s n ,ỹ(s), t(s)) = (s n , y(s), t(s)). Then, by the arc-wise analyticity of Φ, the map q(s, t) = Φ(s n ,ỹ(s), t) is analytic in both s and t, and its complexification q(u, t) is a complex analytic wing containing q(s).
A similar property holds for each component Σ k of the singular locus.
The map for v small, parameterizes a regular wedge W q . The inverse image π −1 (W q ) ∩ X is a finite union of nicely-situated quasi-wings and one of them contains p(s).
Corollary 7.8 (Existence of quasi-wings II). Suppose that p(s) = (s n , y(s), z(s), t(s)) is a real analytic arc in X and not contained in the singular locus Σ f . Then, for b 0 small and generic, p(s) belongs to a quasi-wing in the coordinates x, (Here by generic we mean in {b ∈ C; |b| < ε} \ A, where A is finite. Moreover, we show that one may choose ε > 0 independent of p(s).) Proof. Recall that l i := ord s dist(π(p(s)), π(C i )),l k := ord s dist(π(p(s)), π(Σ k )).
If alll i = l i ≤ m i , i ∈ I C ,l k = l k , k ∈ I Σ then the result follows from Proposition 7.7. Nevertheless, whether this is satisfied or not, it follows from Lemma 7.5 that l i = ord s (y(s) − y i (s, t(s))).
We denote π b (x, y, z, t) := (x, y − bz, t) and by C i,b the associated polar set. By Transversality Assumption X is Zariski equisingular with respect to π b for b sufficiently small (that defines ε). We claim that ifl i > l i and l i ≤ m i then the order ord s dist(π b (p(s)), π b (C i )) = l i , for b = 0. Indeed, otherwise this order is strictly bigger than l i and then, again by Lemma 7.5, |y(s) − y i (s, t(s)) − b(z(s) − z i (s, t(s)))| ≪ s l i . Byl i > l i we have |y(s) − y i (s, t(s))| ≪ s l i and therefore |z(s) − z i (s, t(s))| ≪ s l i that contradicts ord s dist(p(s), C i ) = l i . Moreover, we claim that ord s dist(π b (p(s)), π b (C i,b )) = l i , for b = 0 and small. Indeed, by (17), The first summand is of size sl i , the second one of size bs l i , and the third one of size b 2 s m i . Therefore the claim follows for small b = 0 because l i ≤ m i .
If l i > m i then ord s dist(p(s), C i,b ) = m i for b = 0. Therefore, in general, only for finitely many b, one for each C i , we do not have ord s dist(p(s), Finally, by a similar argument, ord s dist(p(s), Σ k ) = ord s dist(π b (p(s)), π b (Σ k )) for all b but one.
Thus the statement follows from Proposition 7.7.

Basic properties of quasi-wings.
Let p(s) be an arc as given in (39) satisfying the assumptions of Proposition 7.7 and let QW be the quasi-wing constructed in the proof of this proposition. Let p(u, v, t) = (q(u, v, t), z(u, v, t)) be its parameterization. Then, by Lemma 7.5,l i = ord s (y(s) − y i (s, t(s))) and dist(p(s), C i ) ∼ dist(p(s), PW i ) ∼ s l i (and recalll i = l i ≥ m i ). We shall show that the distances from QW to PW i and to Σ k are constant, that is, they are of order u l i and u l k respectively. This follows from their construction that uses arc-wise trivializations of [17] and the partial Lipschitz property of these trivializations, see Remark 7.6.
The following proposition extends the conclusion of Lemma 7.5 from the complex analytic wing q(u, t) to the quasiwing QW. Proposition 7.9. Let QW be the quasi-wing containing p(s) given by Proposition 7.7 and let p(u, v, t) = (q(u, v, t), z(u, v, t)) be its parameterization. Then for the polar sets C i parameterized by p i (u, t) and Σ k by p k (u, t), This implies that dist(p(u, v, t), PW i ) ∼ u l i and dist(p(u, v, t), Σ k ) ∼ u l k .
Proof. It would be convenient in the proof to use the constant ε of Definition 4.1 and denote for this constant fixed, i.e. for |b| < ǫ, the polar wedges by PW i,ε and by PW i,ε their closure. We denote by PW ε (and by PW ε ) the union of PW i,ε (respectively of PW i,ε ) and the singular set Σ f . Lemma 7.10.Φ preserves the polar wedges in the following sense. There is a constant L (depending on the Lipschitz constant of Ψ 2 for its partial Lipschitz property, see Remark 7.6) such that Proof. By constructionΦ preserves the polar set and the singular locus. Therefore the lemma follows from the partial Lipschitz property of Ψ 2 and parameterization (15).
Lemma 7.11. The following holds: Proof. Let l = max i∈I l i . First for fixed ε > 0 we show that It is clear that this distance is , this already holds after the projection π. We show the opposite inequality.
Fix s 0 > 0. By Lemma 7.5 dist(q 0 (s 0 ), π(PW ε ) ∩ {t = 0, s = s 0 }) ∼ s l 0 . Let c(s 0 ) be such that this distance equals exactly c(s 0 )s l 0 and let q min (s 0 ) be one of the points in π(PW ε ) ∩ {t = 0, s = s 0 } realizing this distance. Let τ be the lift of the segment joining q 0 (s 0 ) = π(p 0 (s 0 )) and q min (s 0 ). Since τ is in the complement of PW ǫ (except if its endpoint is in Σ f ), by the boundness of partial derivatives, c.f. the argument of the proof of Proposition 7.3, its length is comparable to the length of the segment, that is s l 0 . Denote by p min (s 0 ) the other endpoint of this lift, so that q min (s 0 ) = π(p min (s 0 )). Since Ψ 2 is partially Lipschitz andΦ preserves the complement of PW ǫ , see Lemma 7.10, we have for small t dist(Φ(p 0 (s 0 ), t),Φ(p min (s 0 ), t)) s l 0 .
Since the distance c(s 0 )s l 0 is a subanalytic function we may suppose, by a choice of q min (s 0 ), that also q min (s 0 ) and p min (s 0 ) are subanalytic in s 0 .
There are three cases to consider p min (s 0 ) ∈ PW ε \ Σ f , p min (s 0 ) ∈ Σ f , and p min (s 0 ) / ∈ PW ε . If p min (s 0 ) is in PW ε \Σ f then, sinceΦ preserves the polar set, so isΦ(p min (s 0 ), t), and the claim follows from (45). A similar argument applies if p min (s 0 ) ∈ Σ f . If p min (s 0 ) / ∈ PW ε then there is another point in π −1 (q min (s 0 )) that is in PW ε . Suppose that it is in PW j,ǫ and denote it by p j (s 0 ). By the assumptions l j =l j =l = l and by the partial Lipschitz property the magnitude of dist(Φ(p j (s 0 ), t),Φ(p min (s 0 ), t)) is independent of t, say ∼ s α 0 . If α ≥ l then (44) follows from (45). If α < l then dist(Φ(p j (s 0 ), t),Φ(p min (s 0 ), t)) ∼ dist(PW j ,Φ(p min (s 0 ), t)) and therefore dist(Φ(p j (s 0 ), t),Φ(p 0 (s 0 ), t)) ∼ dist(PW j ,Φ(p 0 (s 0 ), t)). But, by assumption on the curve p(s) =Φ(p 0 (s), t(s)), dist(Φ(p j (s 0 ), t(s 0 )),Φ(p min (s 0 ), t(s 0 ))) ≤ dist(Φ(p j (s 0 ), t(s 0 )), p(s 0 )) + dist(p(s 0 ),Φ(p min (s 0 ), t(s 0 ))) ≤ Cs l 0 , for a universal constant C. This shows that the case α < l is impossible. Now we show that (44) implies the claim of lemma. Again, it is enough to show since the opposite inequality is already known for the sets projected by π. Firstly, the distance on the left-hand side of (44) has to be attained on one of PW j,ε or Σ k . Suppose, for simplicity, that it is PW j,ε . Then l = l j , that implies the claim of lemma for i = j. By the above there is a curve p j (s) ∈ PW j ∩ {t = 0} such that Let i = j. Then l i ≤ l j and To complete the proof we note that dist(Φ(p j (s), t), PW i ) ∼ s k ij and k ij is also the order of contact between the discriminant branches ∆ i and ∆ j . If l i < l j then dist(q(s, t), ∆ i ) ∼ dist(∆ i , ∆ j ) ∼ s k i,j , and by (43), l i =l i = k i,j .
If l i = l j then k i,j < l i = l j is impossible. Thus k i,j ≥ l j and the RHS of (47) is bounded by s l i = s l j as claimed. This ends the proof of Lemma 7.11.
To show Proposition 7.9 we note that (y i (u, t) − y(u, t)) ∼ u l i by Lemma 7.5 and z i (s, t) − z(s, t) is divisible by s l i for s real and hence z i (u, t) − z(u, t) is divisible by u l i . 24 Corollary 7.12. Under the assumption of Proposition 7.9, we have (y i (u, t) − y(u, t)) ∼ u l i and z i (u, t) − z(u, t) = O(u l i ) for all i ∈ I = I C ∪ I Σ .

Lipschitz vector fields on quasi-wings
Let the quasi-wings QW τ over a fixed regular wedge W q parameterized by (35) be given by We consider such parameterizations for u in an allowable sector Ξ = Ξ I = {u ∈ C; arg u ∈ I}. Then we may write these parameterizations in terms of t, x, v assuming implicitly that we work over a sector Ξ and, moreover, that z τ (x, v, t) is a single valued functions. Again, in order to avoid heavy notation we do not use special symbols for the restriction of a quasi-wing parameterization to an allowable sector.
Even if the parameterizations of quasi-wings carry many similarities to the parameterizations of polar wedges, the boundness of partial derivatives (the property (4) of the beginning of the previous section) is opposite to the very definition of polar set, the vertical tangent versus the horizontal tangents. This boundness and the fact that the projection π restricted to a quasi-wing is bi-Lipschitz make the work with the Lipschitz geometry of quasi-wings in principle simpler.
Proposition 8.1. For all τ and for all x 1 , x 2 , v 1 , v 2 , t 1 , t 2 sufficiently small For every pair of parameterizations p τ , p ν where r τ ν are given by (37).
By Proposition 8.1, h τ (x, v; t) defines a Lipschitz function on the quasiwing QW τ if and only if Given two nicely-situated quasi-wings. Let h be a function defined on a subset of QW τ ∪ QW ν whose restrictions to QW τ , QW ν we denote by h τ (x, v, t) = h • p τ , h ν (x, v, t) = h • p ν respectively. Then, after Proposition 8.1, h is Lipschitz iff so are its restrictions h τ , h ν and Proposition 8.2. The vector fields given on QW τ ∪QW ν by p k * (v), k = τ, ν, where v are ∂ ∂t , x ∂ ∂x , or ∂ ∂v , are Lipschitz. 25 This result is analogous to Proposition 5.1. The only difference comes from the fact that b ∂ ∂b is replaced by ∂ ∂v , since we do not require the vector field to be tangent to the set given by v = 0. The proof we sketch below is simpler that the one of Proposition 5.1 thanks to the mentioned above bi-Lipschitz property.
Proof. First we check that the partial derivatives ∂ ∂t , x ∂ ∂x , ∂ ∂y of the coefficients of these vector fields are bounded. Since nx ∂ ∂x = u ∂ ∂u and ∂ ∂y = u −l ∂ ∂v for the latter two it is more convenient to check that u ∂ ∂u is bounded by x = u n , and ∂ ∂v is bounded by u l . Then the claim follows from the facts that y(u, v, t), z τ (u, v, t) are analytic and divisible by u n , and ∂ ∂v y(u, v, t), ∂ ∂v z τ (u, v, t) are divisible by u l . This shows that these vector fields are Lipschitz on each wing QW τ , QW ν .
To obtain the Lipschitz property between the points of QW τ and QW ν we use a similar argument. Namely, we use formula (37) to show that ∂ ∂t (z τ − z ν ), ∂ ∂u (z τ − z ν ), ∂ ∂v (z τ −z ν ) are bounded (up to a constant) by z τ −z ν , and we complete using formulas (49) and (50).
Let p τ, * (w) be a vector field on QW τ , where We always suppose the vector field p τ, * (w) is well defined on QW τ , that is independent of v if x = 0, and it is stratified, that is tangent to T . The independence on v if x = 0 implies that both α(0, v, t) and β(0, v, t) are independent on v, and the tangency to T assures that in fact β(0, v, t) = 0. Note also that p i * ( ∂ ∂v ) is always zero on x = 0.

Extension of Lipschitz vector fields from PW to an arc in its complement
Suppose we are given a stratified Lipschitz vector field w on S. By the first part of the proof of Theorem 2.1, Section 6, we may extend it to a Lipschitz vector field, still called w, onto PW. In this section we show how to extend it further on the image of a real analytic arc germ p(s) of the form (39) not included in PW. For this we use Corollary 7.8 to embedd p(s) in a quasi-wing QW and extend the vector field from PW to QW. The latter extension is explained in Proposition 9.4. In the process we encounter two problems, discussed below, related to the fact that the construction of Corollary 7.8 gives a quasi-wing after a linear change of coordinates.
If PW i is a polar wedge in the original system of coordinates then we may choose the corresponding polar wedge in the new system of coordinates x, y − b 0 z, z, t, denoted by PW i,b 0 , included in PW i , but we cannot assume that it contains the spine of PW i , that is C i . Therefore, if we extend w|PW i,b 0 to QW using Proposition 9.4, a priori there is no guarantee that the obtained vector field is Lipschitz on PW i ∪ QW. To guarantee it we show that the distance from the arc p(s), and hence from the whole quasi-wing QW, to PW i and to PW i,b 0 are of the same orders. This will follow from Proposition 9.1.
The second problem comes from the fact that the description of stratified Lipschitz vector fields on a polar wedge, given in the conditions 1)-3) of Proposition 5.2, change slightly when we pass from PW i to PW i,b 0 , if PW i,b 0 does not contain C i . Therefore to show Proposition 9.4 one should not use the condition 3). To solve this problem we replace in the proof of Proposition 9.4 the condition 3) by a slightly weaker condition 3') that is satisfied on PW i,b 0 . 9.1. Distance to polar wedges. For the proof of Proposition 9.1 we need the following lemma. By (54) we mean that there is δ > 0 such that dist(γ(s), C i ) ≤ (x(s), y(s), z(s)) δ+m i /n .
We do not claim in the lemma that γ(s) has to belong to the polar wedge containing C i , that is PW i .
Proof. We write the proof in the non-parameterized case. The proof in the parameterized case is similar.
We may suppose that the arc γ is of the form γ(s) = (s n , y(s), z(s)) with y(s) = O(s n ), z(s) = O(s n ) and note that in this case dist(γ(s), C i ) ∼ |y(s) − y i (s)| + |z(s) − z i (s)|. Therefore, by (54), |y(s) − y i (s)| = o(s m i ) and |z(s) − z i (s)| = o(s m i ). Complexify γ by setting γ(u) = (u n , y(u), z(u)). Then, as in the proof of Corollary 7.8, we construct a quasi-wing QW containing γ by changing the system of coordinates, that is replacing y by Y = y − b 0 z, for b 0 sufficiently generic. In this new system of coordinates x, Y, z, t (we do not change the parameter b) the parameterizations of PW i and QW are, x = u n and, respectively, To see that the exponent in the latter formula is m i note that, if we denote the polar set in PW i in the new system of coordinates by C i,b 0 then dist(γ(s), C i,b 0 ) ∼ s m i and we conclude by Corollary 7.12. Now we argue as follows. By Proposition 7.3 the polar wedge PW i and the quasi-wing QW are disjoint (if the constants defining them are small). But if the limit of tangent spaces to X along C i and along γ do not coincide then the implicit function theorem forces PW i and QW to intersect along a curve and therefore this case cannot happen. This is the geometric idea behind the computation below.
Note that (54) implies that, for the old system of coordinates, l i > m i . Therefore the intersection PW i ∩ QW, defined by Y i (u, b) = Y (u, v) and z i (u, b) = z(u, v), is given by the following system of equations There are two cases: (i) Suppose the jacobian determinant of the LHS of (57), with respect to variables b, v is nonzero at u = b = v = 0. Then, by the Implicit Function Theorem there is a solution (b, v) = (b(u), v(u)) of (57), such that b(u) → 0 and v(u) → 0 as u → 0. Then the intersection PW i ∩ QW is the curve parameterized by u: (u n , Y i (u, b(u)), z i (u, b(u))) = (u n , Y (u, v(u)), z(u, v(u))). Therefore, by Proposition 7.3, this case cannot happen. (ii) Suppose that the jacobian determinant of the LHS of (57) vanishes at u = b = v = 0. Then the partial derivatives that are both non-zero at u = b = v = 0, are proportional. This means that the limits of tangent spaces to X along C i , i. e. at (u n , y i (u, 0), z i (u, 0)) as u → 0, and at γ(u) as u → 0, coincide. This limit is transverse to H = {x = 0} since H is not a limit of tangent spaces by the Transversality Assumptions. Hence the tangent spaces to X at γ(u), for small u, contain vectors of the form (0, b, 1) with b → 0 as u → 0. This shows that γ ∈ PW (but not necessarily γ ∈ PW i ). The proof of lemma is now complete.
Proof of Proposition 9.1. The proof is the same in the parameterized and the nonparameterized case. We may suppose again that γ(s) = (s n , y(s), z(s)) with y(s) = O(s n ), z(s) = O(s n ).
If dist(γ(s), S) = dist(γ(s), C i ) then the conclusion for j = i follows directly from Lemma 9.3. Then consider j = i. If the conclusion is not satisfied then In particular, m i > m j , and therefore by Remark 3.7, k ij ≤ m j < m i . But this is impossible since then s m j s k ij ≃ dist(p i (s), p j (s)) dist(C j , γ(s)) + dist(C i , γ(s)) ≪ s m j , where p i , p j denote parameterizations of C i and C j respectively. This ends the proof in this case.
Fix a polar wedge PW i (or Σ k ) closest to QW and parameterized by Recall after Definition 7.4 that m i ≥ l = l i and then by Corollary 7.12 Our goal is to extend any Lipschitz stratified vector field on PW i onto QW. Recall, after Proposition 5.2, that if p i * (α ∂ ∂t + β ∂ ∂x + δ ∂ ∂b ) is Lipschitz stratified then α, β, and δ satisfy the conditions 1)-3) of Proposition 5.2. In what follows we use only a weaker version of condition 3) that is, see Remark 9.5 for explanation, 3') |δ| is bounded and δx m/n satisfies (26).
We note that by (58) and m i ≥ l, a vector field is Lipschitz on PW i ∪ QW if and only if it is Lipschitz on each PW i and QW and it is Lipschitz on the union of the images of two arcs p(u, t) and p i (u, t).
Proof. By Proposition 8.3, p * (α 0 ∂ ∂t + β 0 ∂ ∂x ) is Lipschitz on QW. To show that both vector fields define a Lipschitz vector field on PW i ∪ QW it suffices to show that, taking b = 0 and v = 0 we have: The items (1)-(4) follow from (58) and (5) follows from m i ≥ l i . Remark 9.5. Since in the above proof we only used the condition 3') we can apply Proposition 9.4 to the quasi-wings constructed in Corollary 7.8, that is after a change of coordinates to x, Y b 0 , z, t, where Y b 0 := y − b 0 z, that corresponds to a shift in b.

Proof of Theorem 2.1. Part II
We complete the proof of Theorem 2.1. Let γ(s), γ ′ (s), s ∈ [0, ε), be two real analytic arcs in X . We want to show that any stratified Lipschitz vector field v defined on the union of S and γ extends to γ ′ as stated in the valuative criterion, see the next section. We consider two cases. Case 1. dist(γ(s), γ ′ (s)) dist(γ ′ (s), S). Then it is enough to extend v| S to a Lipschitz vector field on S ∪ γ ′ , since then such an extension defines a Lipschitz vector field on S ∪γ(s) ∪γ ′ (s) for every s sufficiently small, with the Lipschitz constant independent of s. Case 2. dist(γ(s), γ ′ (s)) ≪ dist(γ ′ (s), S). Then it suffices to extend v from γ to a Lipschitz vector field on γ ∪ γ ′ . Note that we may suppose that on both arcs γ, γ ′ we have that y = O(x), z = O(x), that is, they are in the form (32). Indeed, by Transversality Assumption the variable z restricted to an arc in X cannot dominate x and y, that is In this case we change the local coordinate system to (X a , y, z, t) = (x − ay, y, z, t), for a = 0 and small. This is a change of coordinates in the target of the projection (x, y, z, t) → (x, y, t) and affects neither the discriminant locus nor Zariski's Equisingularity.
To make the proof more precise we will use the constant ε of Definition 4.1 and denote thus defined the union of polar wedges and the singular set by PW ε . If both γ(s), γ ′ (s) belong to PW ε then the claim follows from the first part of the proof, Section 6.
In Case 1, given a stratified Lipschitz vector field v on S we extend it on γ ′ . By Proposition 9.1 we may suppose dist(γ(s), C j ) s m j for every j, and therefore, for b small, say b ≤ ε, dist(γ(s), C j ) ∼ dist(γ(s), C j,b ), where C j,b denotes the polar set in PW j after the change of coordinates to x, Y b 0 = y − b 0 z, z, t. Then we proceed as follows. First we extend v to a Lipschitz vector field on PW ε/2 and use Corollary 7.8 to embedd γ ′ in a quasi-wing in this new system of coordinates for a b 0 ≤ ε/2. Thus there exists a quasi-wing QW containing γ ′ and, moreover, dist(γ ′ (s), S) = dist(π b 0 (γ ′ (s)), ∆ b 0 ) ∼ s l , where l = max{max l i , max r k } and ∆ b 0 denotes the discriminant π b 0 . Then there is a Lipschitz extension of v to QW by Proposition 9.4.

Valuative criterion on extension of Lipschitz vector fields
The purpose of this section is to give a precise statement of a valuative criterion on extension of Lipschitz vector fields. In this criterion we formalise our strategy of checking the conditions (i) and (ii) of Proposition 2.4 along real analytic arcs.
Let us consider the following more general set-up. Let X be a locally closed subanalytic subset R n with a filtration F = (X j ) j=l,...,d by closed subanalytic subsets X = X d ⊃ X d−1 ⊃ · · · ⊃ X l = ∅, such that for every j = l, . . . , d,X j = X j \ X j−1 is either empty or a real analytic submanifold of pure dimension j. Here we mean X l−1 = ∅. Note that F induces a stratification of X by taking the connected components of everyX j as strata. By a stratified Lipschitz vector field (SLVF for short) we mean a Lipschitz vector field defined on a subset of X and tangent to the strata.
We say that F induces a Lipschitz stratification at p ∈ X if there is an open neighbourhood U of p such that F restricted to U induces a Lipschitz stratification of X ∩ U. Proof. We first recall the notions of a chain and Mostowski's Conditions. We follow the approach of [13] simplifying a little bit the notation and exposition. For slightly different but equivalent conditions see [9,14]. One can simplify the proof below by using directly the valuative criteria of [6] but we prefer to give a self-contained proof based on elementary computations given in the proofs of Proposition 1.2 and 1.5 of [13].
Fix c > 1. A chain (more exactly, a c-chain) for a point q ∈X j is a strictly decreasing sequence of indices j = j 1 , j 2 , . . . , j r = l and a sequence of points q m ∈ X jm such that q 1 = q and j m is the greatest integer for which dist(q, X k ) ≥ 2c 2 dist(q, X jm ) for all k < j m |q − q m | ≤ c dist(q, X jm ).
Let P q : R n → T qX j denote the orthogonal projection onto the tangent space and P ⊥ q = I − P q the orthogonal projection onto the normal space T ⊥ qX j . We say that F satisfies Mostowski's Conditions if there is a constant C > 0 such that for all chains {q m } m=1,...,r and all 2 ≤ k ≤ r: If, further, q ′ ∈X j and |q − q ′ | ≤ ( 1 2c ) dist(q, X j−1 ) then |(P q − P q ′ )P q 2 · · · P q k | ≤ C|q − q ′ |/ dist(q, X j k −1 ), (M2) in particular, where dist(·, ∅) ≡ 1.
By Proposition 1.5 of [13], F induces a Lipschitz stratification if and only if any of two equivalent conditions (i) and (ii) of Proposition 2.4 holds. In particular the definition of Mostowski's stratification is independent of the choice of the constant c > 1 used to define the chains.
Clearly by Proposition 2.4 a Lipschitz stratification satisfies LVE condition at any point of X.
Suppose that F satisfies LVE condition at p. We show by induction on j that F induces a Lipschitz statification of X j at p, the case j = l being obvious because X l is nonsingular. Thus we suppose it for X j−1 and prove for X j . Suppose the latter does not hold. Then by a fairly straightforward application of the curve selection lemma there are real analytic arcs q m (s) : [0, ε) → X jm , m = 1, . . . , r, j 1 = j, at p, that are c-chains of q(s) = q 1 (s) for s = 0, and possibly another arc q ′ (s) : [0, ε) → X j satisfying |q(s) − q ′ (s)| ≤ ( 1 2c ) dist(q(s), X j−1 ) for s = 0, for which one of the conditions (M1),(M2) fails, that is it holds with the constant C(s) → ∞ as s → 0. Indeed, it follows from Lemma 6.2 of [9], that is stated in the complex analytic set-up, or from the valuative criteria of [6], where the authors even managed to get rid of the constant c defining the chains.
We will show that the existence of such arcs contradicts LVE condition. We may assume that the index k, given by the length of the expression on the left-hand side of (M1),(M2), for which one of these conditions fails is minimal. Suppose that this is the condition (M1). Let us then put γ ′ (s) := q(s) and γ(s) := q 2 (s). Then adapting the proofs of Propositions 1.2 and 1.5 of [13] and using LVE condition we show that there is a constant C > 0, independent of s, such that (M1) holds along the family of arcs q m , m = 1, . . . , k, that gives a contradiction.
Note that if k = 2 the first term of the RHS of the first inequality does not appear, otherwise everything is the same. Since w(q k (s)) = dist(q k (s), X j k −1 )P q k (s) v we get, by property (3) of the chains, |P ⊥ q 1 (s) P q 2 (s) · · · P q k (s) v| ≤ C ′ |q(s) − q 2 (s)|/ dist(q(s), X j k −1 ). Applying the above to a finite set of v from an orthonormal basis of V 0 , and taking into account that |P q k (s) v − v| ≤ C|q k (s)| → 0, as s → 0, we show that (M1) holds along this family of arcs contrary to our assumptions. A similar argument, based on the second part of the proof of Proposition 1.5 of [13] applies to the condition (M2). This ends the proof.
Remark 11.4. Proposition 11.3 holds in a more general o-minimal set-up when one assumes every X j to be definable, everyX j to be a C 2 submanifold, and the arcs to be continuous and definable. One can also restrict the LVE condition, Definition 11.1, to definable vector fields, because the extension of Lipschitz vector fields construction of Proposition 1.2 of [13] preserves the definability, see Remark 1.4 of [14].