BI-LIPSCHITZ TRIVIAL QUASI-HOMOGENEOUS STRATIFICATIONS
DWI JUNIATI AND GUILLAUME VALETTE
Abstract. We call a stratification quasi-homogeneous when it is invariant by a certain
class of action. The paper gives criteria for a quasi-homogeneous stratification to fulfill
the (w) condition of Kuo-Verdier or to be bi-Lipschitz trivial. We also study the variation
of the volume of some quasi-homogeneous families of germs.
1. Introduction and known results
The notion of stratification is one of the fundamental concepts in algebraic geometry
and singularity theory. The idea to decompose a singular topological space X into non
singular parts which we call the strata, satisfying some regularity conditions, goes back to
Whitney’s and Thom’s work.
One of the most famous regularity conditions for a stratification is the (w) condition of
Kuo-Verdier. It is defined in the following way. First, given E and F two vector subspaces
of Rn we define δ(E; F ) =
sup d(u; F ) where k . k is the Euclidian norm and d the
u∈E,kuk=1
Euclidian distance. Then:
Definition 1.1. (Cf. [V] and [K]) We recall that a pair (X, Y ) of submanifolds of Rn ,
with Y ⊂ X − X, is said to be a (w) stratification (or to satisfy the (w)-condition)
at y0 ∈ Y if there is a real constant C > 0 and a neighborhood U of y0 such that
δ(Ty Y, Tx X) ≤ C k x − y k,
for all y ∈ U ∩ Y and all x ∈ U ∩ X.
Using Hironaka’s resolution of singularities, Verdier established that every analytic variety admits a (w) stratification. For differentiable stratifications, the (w) condition implies
local topological triviality along strata [V].
T. Mostowski introduced the notion of Lipschitz stratification which enabled him to
give a bi-Lipschitz version of the Thom-Mather isotopy theorem and proved its existence
for complex analytic sets. His work has then been extended to the real setting by A.
Parusiński. In [Va2], the second author has given a semi-algebraic bi-Lipschitz isotopy
theorem for sets which are definable in an o-minimal structure. In [Bi 1-2] Lev Birbrair
has investigated the problem of classification of semi-algebraic sets, up to bi-Lipschitz
equivalence. For a classification of (w)-stratifications and Lipschitz stratifications for the
surfaces y a = z b xc + xd , where a, b, c, d are positive integers, see [N],[J], and [J-T].
In stratification theory, quasi-homogeneous spaces are often studied. S. Koike in [Koi1]
and [Koi2], studied the modified Nash triviality and the case of quasi-homogeneous families
with isolated singularities.
Date: July 13, 2006.
1991 Mathematics Subject Classification. 14D06, 57N80, 58K25.
Key words and phrases. stratification,Whitney stratification, Lipschitz stratification, Lipschitz triviality.
1
2
DWI JUNIATI AND GUILLAUME VALETTE
T. Fukui and L. Paunescu [FP] found some conditions on the equations defining the
strata to show that a given family is topologically trivial. They gave a weighted version
of (w) regularity and of Kuo’s ratio test, which provides weaker conditions for topological
stability. After that, Bohao Sun [Sun] took these methods to generalize the work of D.
Trotman and L. Wilson [Tr-W] for the (t) condition.
K. Bekka in [B] gives sufficient conditions for a quasi-homogeneous stratification to
satisfy the (c) condition. The problem of bi-Lipschitz stability of quasi-homogeneous
germs is studied in [FR] from the point of view of sufficiency of jets. They give some
interesting explicit criteria for bi-Lipschitz determinacy of quasi-homogeneous germs. Our
approach is similar but we are concerned by stratification theory.
In this paper, we give some conditions for a quasi-homogeneous set stratified by two
strata (X; Y ) to be bi-Lipschitz trivial along Y . We also study the variation of the volume
of quasi-homogeneous stratified families.
Acknowledgement. The first author thanks ICTP-Trieste for support and helpful working conditions. This work was partially done at ICTP when the first author came as an
associate member of ICTP. The second author is grateful to the Jagiellonian University
of Kraków for its hospitality. We thank the referee for his relevant remarks.
First we need some definitions. Throughout this paper C stands for various positive
constants. We fix two integers k and n with k ≤ n.
Definition 1.2. Let V be a R- finite dimensional vector space. A R -action α on V is a
map given by:
α : R \ {0} × V −→ V
(t, x) = (t, (x1 , · · · , xn )) 7→ t.x = (ta1 x1 , · · · , tan xn )
with a1 , a2 , ..., an integers, called the weights of the action α.
Definition 1.3. Let Σ be a stratification of X. Let α be an action defined on X. We say
that Σ is invariant by α if for any nonzero real number t we have t.S ⊆ S, ∀S ∈ Σ.
We also say that Σ is a quasi-homogeneous stratification.
Important: We will always assume that ai ≤ 0, ∀i ≤ k and ai > 0, ∀i > k.
Let α be an action with the weights a = (a1 , · · · , an ). Define the positive continuous
function ρα by:
ρα : R n → R
ρα (x) = ρα ( (x1 , . . . , xn ) ) =
X
m
| xi | ai
1
m
,
i>k
where m = l.c.m. of {ai | k < i ≤ n}.
As ai ≥ 0 for i > k, we see that ρα is a differentiable function outside Rk ×{0}. Moreover
clearly we have:
ρα (t.x) = | t | ρα (x)
∀t ∈ R.
BI-LIPSCHITZ TRIVIAL QUASI-HOMOGENEOUS STRATIFICATIONS
3
2. Quasi-homogeneous (w) stratifications
Now, we are going to give some sufficient conditions for a couple of strata to satisfy the
(w) condition in the case of a quasi-homogeneous stratifications. This estimation will be
also used in the next section to have a criterion for the condition (w) with exponent k.
From now on we will use the following notations:
SX = Sup ai ;
i>k
IX = Inf ai
and
i>k
SY = Sup ai .
i≤k
where ai ≥ 0 for all i > k and ai < 0 for all i ≤ k.
Lemma 2.1. Let (X, Y) be a couple of strata, with Y ⊂ X − X, and Y = Rk × 0,
X ⊆ Rn , X ∪ Y locally closed. Let α be an action defined on this stratification. If (X, Y )
is invariant by α there exist ε, a positive constant C and a neighborhood of the origin
U ⊂ Y such that δ(Y, Tq X) < 1, for all q ∈ U × Rn−k satisfying ρα (q) = ε we have:
δ(Y, Tq X) ≤ C ρα (q)IX −SY .
Proof. By hypothesis we can find a real ε such that:
(2.1)
δ(Y, Tq X) ≤ c < 1,
if ρα (q) = ε in a relatively compact neighborhood of the origin U ⊂ Y .
Then for each point p such that ρα (p) = ε, we can find u in Tp X such that π(u) = ej =
(0, ..., 0, 1, 0, ..., 0), where the j th coordinate is = 1, with j ≤ k (where π is the projection
onto Y) and |u| is bounded by a constant C independent of p.
Let q ∈ X and t = ραε(q) . Then ρα (t.q) = ε, by definition of ρα . Note that as ai ≤ 0
for i ∈ {1, . . . , k} the point 1t .q stays in a relatively compact neighborhood of the origin.
Therefore (2.1) is fulfilled by 1t .q.
Hence we can choose u = (u1 , ..., un ) ∈ Tt.q X satisfying π(u) = ej and |u| ≤ C.
Remark that 1t .u ∈ Tq X because X is invariant by α. Note that
δ(Y, Tq X) ≤ max
1≤j≤k
| 1t .u − 1t .ej |
( 1t )aj
and
| 0, ..., 0, ( 1t )ak+1 uk+1 , ..., ( 1t )an un |
| 1t .u − 1t .ej |
=
( 1t )aj
( 1t )aj
1
1
1
= ( )IX −aj 0, ..., 0, ( )ak+1 −IX uk+1 , ..., ( )an −IX un t
t
t
1 IX −aj
≤ C( )
t
1 IX −SY
≤ C( )
t
1
(because aj ≤ 0 and is small for q in the neighborhood of zero).
t
Hence we can conclude that δ(Y, Tq X) ≤ C ρα (q)IX −SY .
Theorem 2.2. In the situation of Lemma 2.1, if IX − SY ≥ SX , then δ(Y, Tq X) ≤
S
C ρα (q) X on a neighborhood of the origin. In particular the pair (X, Y ) is a (w)
stratification.
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DWI JUNIATI AND GUILLAUME VALETTE
Proof. If IX − SY ≥ SX , then for q sufficiently near the stratum Y ,
ρα (q)IX −SY ≤ ρα (q)SX .
So, using Lemma 2.1:
S
δ(Y, Tq X) ≤ C ρα (q) X
But there exists a constant C such that :
S
X
m X
SX
m
ai
ρα (q)
=
| xi |
i>k
≤ C
X
≤ C
X
| xi |
SX
ai
i>k
| xi | .
i>k
Hence (X; Y ) satisfies the (w) condition at the origin.
In particular, this implies that X is topologically trivial along Y .
Remark 2.1. The criterion given in Theorem 2.2 may be generalized to more than two
strata {Y, X1 , . . . , Xm }. Nevertheless one has to assume that the couples (Xi ; Xj ) are (w)
regular in order to deduce that the couples (Y ; Xi ) are as well.
Example 2.1. Let V = {(x, y, u, v) ⊂ R4 | y 4 = x5 u2 v 4 + x2 }, then Sing V = {0} ×
{0} × R2 .
Stratify V with X = Reg V and Y = Sing V . Define an action α by t.(x, y, u, v) =
(t4 x, t2 y, t−2 u, t−2 v).
Then, we have SX = 4, IX = 2, SY = −2. The pair (X; Y ) is invariant by α and the
condition of Theorem 2.2, in this case is IX − SY ≥ SX . We know that: δ(Y, Tq X) = 1 ⇔
∂f
∂f
4
5 2 4
2
∂x = 0 and ∂y = 0, with f = y − x u v − x . We have:
∂f
∂f
= 0 ⇒ y = 0, and
= 0 ⇒ x = 0.
∂y
∂x
This means that there exists a positive constant ε such that δ(Y, Tq X) < 1, for all q such
that ρα (q) = ε. Hence the pair of (X, Y ) is a (w) regular stratification.
We end this section by a result about the volume of quasi-homogeneous families which
is a consequence of the above and of the results in [Va1].
2.1. On the volume of quasi-homogeneous families. It is a well known fact due to
J.-M. Lion and J.-P. Rolin that the volume of a subanalytic subset intersected by a ball of
radius r centered at the origin has a Puiseux expansion in r and ln r. On the other hand
in [Va1] it is proved that if the Kuo-Verdier condition is satisfied ”at higher orders” the
expansion of the volume is preserved when the parameters vary at a corresponding order.
We will use the results of the previous section about quasi homogeneous-stratified families
to get some sufficient criteria for stability of the expansion of the volume of the fibers of
such families.
Let us write Ay for the germ of A at y ∈ A. Given an analytic subset of Rn and y ∈ Rn
we will set
ψ(Ay ; r) = Hl (A ∩ B(y; r))
where l is the Hausdorff dimension of the germ of A at y and Hl the l dimensional Hausdorff
measure. As we have said this function has a Puiseux expansion in r and ln r.
BI-LIPSCHITZ TRIVIAL QUASI-HOMOGENEOUS STRATIFICATIONS
5
Theorem 2.3. Let A be an analytic subset of Rn stratified by two strata (X; Y ), with
Y = Rk × {0}. If IX − SY ≥ kSX , then the expansion of the volume ψ(Ay ; r) is constant
until the order (k + l − 1) along Y (where l = dim A).
In particular if IX − SY ≥ 2SX then the density of A is constant along Y .
Proof. Applying Theorem 2.2 we get
kS
δ(Y, Tq X) ≤ C ρα (q) X
which means that
δ(Y, Tq X) ≤ C|q − π(q)|k
which is the Kuo-Verdier condition with exponent k. By Theorem 3.2.2 of [Va1] we get an
isotopy ht which is a (k − 1) approximation of the identity (see [Va1]), which means that
|ht (q) − q| ≤ crk−1 .
But now by Theorem 3.1.3 of [Va1] we see that the expansion of ψ(Ay ; r) is constant until
the order k + l − 1. The last statement is due to the fact that the term of order l is, by
definition, the density [Kur-Ra].
Remark 2.2. This theorem has been written for simplicity in the two strata case but
actually works in the more general setting of an arbitrary quasi-homogeneous stratification
(topological triviality is not required). In [Va1] are also given some equimultiplicity results
that can be deduced for families fulfilling the hypothesis of Theorem 2.3.
3. Bi-Lipschitz triviality of quasi-homogeneous stratifications
Now we give sufficient conditions on the weights for a set X to be bi-Lipschitz trivial
along Y.
Proposition 3.1. Let (X, Y ) be a (w) regular couple of strata such that Y ⊆ X − X, Y =
Rk × 0, X ∪ Y locally closed. Let α be an action defined on this stratification. If (X, Y )
is invariant by α, then there exist a neighborhood of zero and a constant C such that for
all couples (q, q 0 ) ∈ X × X in this neighborhood such that | ρα (q) − ρα (q 0 ) | < 12 ρα (q) and
for each unit vector v(q) in Tq X normal to Y, we can find a vector v(q 0 ) in Tq0 X, normal
to Y such that
| q − q0 |
| v(q) − v(q 0 ) | ≤ C
.
ρα (q)(2SX −IX )
Proof. On the set V = {q ∈ X | 2ε ≤ ρα (q) ≤ 3ε
2 }, for ε small enough, we can construct
a vector field which fulfills the desired property since we may construct a smooth one (the
(w) condition implies that Y ⊥ is transverse to Tq X).
Let q ∈ X and t =
ε
ρα (q) .
Let v ∈ Tq X, with ||v|| = 1, π(v) = 0.
Let q 0 ∈ X such that | ρα (q 0 ) − ρα (q) | <
1
2 ρα (q).
Then t.q and t.q 0 belong to V . So, we can find a vector w in T 1 .q0 X such that | t.v − w |≤
t
C | t.q − t.q 0 || t.v |.
Let v(q 0 ) = 1t .w and remark that | t.v |≤ Cρα (q)−SX .
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DWI JUNIATI AND GUILLAUME VALETTE
So
1
| v(q) − v(q 0 ) | ≤| v(q) − .w |
t
1 IX
| t.v(q) − w |
≤C
t
since v and w are normal to Y . This implies that:
1 IX
| t.q − t.q 0 || t.v |
| v(q) − v(q 0 ) | ≤ C
t
1 IX −SX
| q − q 0 | (ρα (q))−SX
≤C
t
| q − q0 |
≤C
2S −I .
ρα (q) X X
Proposition 3.2. Let (X, Y ) be a pair of strata invariant by an action α, with X ∪ Y
locally closed. As in Proposition 3.1 suppose that there exists ε such that δ(Y, Tq X) < 1 if
ρα (q) = ε. Then there exist a neighborhood of the origin and a constant C such that for
each couple (q; q 0 ) in X × X and for each vector ej of the canonical basis of Y , there exist
a vector vj in Tq X and a vector uj in Tq0 X such that:
I −S −a
| vj − uj |≤ C | q − q 0 | ρα (q) X X j
I −a
| vj − ej |≤ C ρα (q) X j
Proof. We use the same method as in the proof of the Proposition 3.1.
Let V = {q ∈ X |
Suppose t =
ε
ρα (q) .
ε
2
≤ ρα (q) ≤
Then t.q and
3ε
2
t.q 0
}.
belong to V .
As in the proof of Lemma 2.1, we can choose a vector w of Tt.q X whose norm is bounded
by a constant independent of q and such that π(w) = ej .
Now we have:
1
1
.w
−
.e
t
t j
( 1t )aj
1
≤ C ( )IX −aj
t
I −a
≤ C ρα (q) X j .
The set V is smooth and t.q 0 ∈ V , so we can choose a vector w0 such that π(w0 ) = ej
and | w − w0 |≤ C | t.q − t.q 0 |.
And by a similar computation as in Lemma 2.1 we get:
1
1
0
t .w − t .w | q − q 0 | IX −SX −aj
≤
C
.
(t)aj
ρα (q)
Hence it is enough to set
vi =
1
t .w
(t)aj
and ui =
1
0
t .w
.
(t)aj
BI-LIPSCHITZ TRIVIAL QUASI-HOMOGENEOUS STRATIFICATIONS
7
Theorem 3.3. In the same situation as in the Proposition 3.2, with X ∪ Y locally closed,
if (2IX − SY ) ≥ 2SX , then the stratum X is bi-Lipschitz trivial along Y on a neighborhood
of the origin.
Proof. It is enough to show that on a neighborhood of the origin, we can find a constant
C satisfying :
| (Pq − Pq0 )π | ≤ C | q − q 0 |
for all couple (q, q 0 ), if π is the orthogonal projection on Y . This inequality clearly enables
us to lift a basis of Y to a Lipschitz family of vector fields whose respective flows will
provide the desired trivialization.
We start by the case where (q; q 0 ) verify | ρα (q) − ρα (q 0 ) | ≤
1
2
ρα (q).
Let 1 ≤ j ≤ k. By Proposition 3.2, there exist a vector vj and a vector wj such that:
I −S
| vj − ej | ≤ C ρα (q) X Y
and
I −S −S
| vj − wj | ≤ C | q − q 0 | . ρα (q) X X Y .
By Proposition 3.1, if u ∈ Tq X is a unit vector normal to Y in Tq X, there exists a
vector u0 in Tq0 X also normal to Y such that
| u − u0 | ≤
| q − q0 |
.
ρα (q)2SX −IX
We have 2IX − SY ≥ 2SX , then:
| vj − ej | ≤ C ρα (q)2SX −IX
(3.1)
and
| vj − wj | ≤ C | q − q 0 | ρα (q)SX −IX
(3.2)
From this one may deduce that:
δ(Tq X, Tq0 X) ≤ C
| q − q0 |
.
ρα (q)2SX −IX
| (Pq − Pq0 ) | ≤ C
| q − q0 |
.
ρα (q)2SX −IX
Which amounts to:
(3.3)
Now:
| (Pq − Pq0 )(ei ) | ≤ C | (Pq − Pq0 )(vi − ei ) | + | (Pq − Pq0 )(v) |
≤ C | Pq − Pq0 | . | vi − ei | + | Pq⊥0 (v) |, using (3.1)
| q − q0 |
. ρα (q)2SX −IX + | Pq⊥0 (vi − wi ) |, by (3.3) and vi ∈ Tq X
ρα (q)2SX −IX
≤ C | q − q 0 | + | vi − wi | ( according to (3.2))
≤C
≤ C | q − q0 | .
Now there remains the case where | ρα (q) − ρα (q 0 ) | ≥
have ρα (q 0 ) ≤ 12 ρα (q).
We have ρα (t.q 0 ) ≤
1
2
ρα (t.q). Also ρα (t.q 0 ) ≤ 2ε .
Let µ = dist(Wε ; W 2ε ) where Wε = {x | ρα (q) = ε}.
1
2
ρα (q). Note that in this case we
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DWI JUNIATI AND GUILLAUME VALETTE
It is clear that | t.q − t.q 0 |≥ µ, so we have :
1
| q − q 0 | = | (t.q − t.q 0 ) |
t
1 SX
≥ | | | t.q − t.q 0 |
t
1
≥ µ( )SX
t
≥ µρα (q)SX .
As 2IX − SY ≥ 2SX , we have IX − SY ≥ SX and according to Theorem 2.2
| Pq⊥ π |≤ Cρα (q)SX .
Then:
| (Pq − Pq0 )π | =| (Pq⊥ − Pq⊥0 )π |
≤ | Pq⊥ π | + | Pq⊥0 π |
≤ C ρα (q)SX + Cρα (q 0 )SX
≤ C ρα (q 0 )SX
≤ C | q − q0 | .
which completes the proof.
Example 3.1. Let V = {(x, y, u, v) ⊂ R4 | y 4 = x6 u2 v 4 + x3 }, then Sing V = {0} ×
{0} × R2 .
Stratify V by X = RegV and Y = SingV .
Define an action α by t.(x, y, u, v) = (t4 x, t3 y, t−2 u, t−2 v).
The pair (X, Y ) is invariant by α and the conditions of Theorem 3.3 in this case: 6−(−2) ≥
8 holds. Moreover
δ(Y, Tq X) = 1 ⇔
∂f
=0
∂x
and
∂f
= 0.
∂y
We know that :
δ(Y, Tq X) = 1 ⇔
∂f
∂x
= 0 and
∂f
∂y
= 0, with f = y 4 − x5 u2 v 4 − x2 .
After some computations we get:
∂f
∂f
= 0 ⇒ y = 0, and
= 0 ⇒ x = 0.
∂y
∂x
This means that there exists ε such that δ(Y, Tq X) < c < 1 for all q such that ρα (q) = ε.
Therefore according to Theorem 3.3 the stratification Σ = (X; Y ) of X ∪ Y is bi-Lipschitz
trivial along Y.
Remark 3.1. Theorem 3.3 provides bi-Lipschitz triviality along Y locally at the origin.
The proof actually also establishes it all along Y (in a sufficiently small neighborhood of
Y ), but the obtained Lipschitz constants are local and in general may be bounded only
on compact subsets of Y .
BI-LIPSCHITZ TRIVIAL QUASI-HOMOGENEOUS STRATIFICATIONS
9
The criterion provided by Theorem 3.3 cannot be used for a stratification having more
than two strata. We end this paper by giving a counterexample.
Example 3.2. Let f : R4 → R4 be defined by f (x; y; z; t) = z 3 y 5 − z 8 x2 t2 + x4 . The zero
locus of this polynomial is invariant through the action u.q := (u−2 t; u2 x; uy; uz). We can
consider this set as a family parameterized by the variable t and define Y to be the t axis.
Note that with the above notations we have: 2IX − SY ≥ 2SX .
However we can see that this family is not bi-Lipschitz trivial along Y . Indeed at any
point q := (t0 ; 0; 0; z0 ) with t0 nonzero, the corresponding fiber of the family is a family of
cusps (along the z axis) with order of contact 45 if t0 is zero and 52 if t0 is nonzero. The t0
axis is the set of points at which the set fiber is not a Lipschitz manifold and hence has
to be preserved by any bi-Lipschitz trivialization of the family and the order of contact of
the cusp should be preserved as well.
References
[B]
K. Bekka, Regular Quasi- Homogeneous Stratifications, Stratifications and Topology of Singular
Spaces, Travaux en cours vol. 55, 1997, pp. 1-14.
[Bi1]
L. Birbrair, Local bi-Lipschitz classification of 2-dimensional semialgebraic sets. Houston J.
Math. 25 (1999), no. 3, 453–472.
[Bi2]
Lev Birbrair, Lipschitz geometry of Curves and surfaces definable in O-Minimal structures,
preprint.
[FP]
T. Fukui and L. Paunescu, Stratification theory from the weighted point of view, Canadian
Journal of Mathematics, 53(1), 2001, pp. 73-97.
[FR]
A. Fernandes, M. Ruas, Bi-Lipschitz determinacy of Quasihomogeneous germs, Glasgow Mathematical Journal, Volume 46, Issue 01, Februar 2004.
[J]
D. Juniati, De la régularité Lipschitz des espaces stratifiés, Thèse,Université de Provence, 2002.
[J-T]
D. Juniati and D. J. A. Trotman, Determination of Lipschitz Stratifications for Surfaces y a =
z b xc + xd , Seminaires et Congrés 10, SMF 2005.
[J-T-V] D. Juniati, D. J. A. Trotman and G. Valette, Lipschitz Stratifications and Generic wings, J.
London Math. Soc. (2) 68,2003, pp. 133-147.
[K]
T.-C. Kuo, The ratio test for analytic Whitney stratifications, Proc. of Liverpool Singularities
Symposium I, Lecture Notes in Math. 192, Springer-Verlag, New York, 1971, pp. 141-149.
[Koi1]
S. Koike, Modified Nash triviality of a family of zero-sets of weighted homogeneous polynomial
mappings, Proceedings of International Workshop on Topology and Geometry at Hanoi, Kodai
Journal Math., 17, 1994, pp. 432-437.
[Koi2]
S. Koike, Modified Nash triviality theorem for a family of zero-sets of weighted homogeneous
polynomial mappings, Journal Math. Soc. Japan Vol. 49, No.3 , 1997, pp. 617-631.
[Kur-Ra] K. Kurdyka and G. Raby, Densité des ensemble sous-analytiques, Ann. Inst. Fourier, Grenoble,
39, 1989, pp. 753-771.
[M]
T. Mostowski, Lipschitz equisingularity, Dissertationes Math., CC XLIII, 1985, PWN, Warsawa.
[N]
L. Noirel, Plongements sous-analytiques d’espaces stratifiés de Thom-Mather, Thèse, Université
de Provence, 1996.
[P]
A. Parusinski, Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier, Grenoble; Vol. 38,
1988, no.4, pp. 189-213.
[Sun]
Bohao Sun, Stratifications and sufficiency of weighted jets, thesis, University of Hawaii at Manoa,
Honolulu, 1996.
[Tr-W] D. Trotman and L. Wilson, Stratifications and finite determinacy, Proc. London Math. Soc. (3)
78, 1999, no. 2, pp.334–368.
[V]
J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36, 1976,
pp. 295-312.
[Va1]
G. Valette, Volume and multiplicities of real analytic sets, Ann. Pol. Math. 87 (2005), 265–276.
[Va2]
G. Valette, Lipschitz triangulations. Illinois J. of Math., Vol. 49, No. 3, Fall 2005, 953-979.
(D. Juniati) Department of Mathematics, Universitas Negeri Surabaya (UNESA), Indonesia
E-mail address: dwi [email protected]
10
DWI JUNIATI AND GUILLAUME VALETTE
(G. Valette) Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, 30 059 Kraków,
Poland
E-mail address: [email protected]