Bất đẳng thức Lojasiewicz, tương đương topo và đa diện Newton (LA tiến sĩ)

MINISTRY OF EDUCATION AND TRAINING
DALAT UNIVERSITY

LOJASIEWICZ INEQUALITIES, TOPOLOGICAL
EQUIVALENCES AND NEWTON POLYHEDRA
Speciality:

Mathematical Analysis

Speciality code: 62.46.01.02

A THESIS
SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

DALAT, 2017


MINISTRY OF EDUCATION AND TRAINING
DALAT UNIVERSITY

BUI NGUYEN THAO NGUYEN

LOJASIEWICZ INEQUALITIES, TOPOLOGICAL
EQUIVALENCES AND NEWTON POLYHEDRA
Speciality:

Mathematical Analysis

Speciality code: 62.46.01.02

A THESIS
SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS
Supervisors:
1. Assoc. Prof. Pham Tien Son
2. Dr. Dinh Si Tiep

DALAT, 2017


Declaration of Authorship
I, B`
ui Nguyˆ˜en Tha˙’o Nguyˆen, declare that this thesis titled, “LOJASIEWICZ INEQUALITIES, TOPOLOGICAL EQUIVALENCES AND NEWTON POLYHEDRA”
and the work presented in it are my own. I confirm that:
• This work was done wholly or mainly while in candidature for a research degree
at this University.
• Where any part of this thesis has previously been submitted for a degree or any
other qualification at this University or any other institution, this has been clearly
stated.
• Where I have consulted the published work of others, this is always clearly attributed.
• Where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this thesis is entirely my own work.
• I have acknowledged all main sources of help.
• Where the thesis is based on work done by myself jointly with others, I have made
clear exactly what was done by others and what I have contributed myself.
Signed:

Date:

i


Abstract
LOJASIEWICZ INEQUALITIES, TOPOLOGICAL EQUIVALENCES
AND NEWTON POLYHEDRA

by B`
ui Nguyˆ˜en Tha˙’o Nguyˆen
The goals of this thesis are to study Lojasiewicz inequalities and topological equivalences (local and at infinity) for a class of functions satisfying non-degenerate conditions.
Singularity Theory and Semi-algebraic Geometry are main tools for our study.
Our main results include:
- Establishing a formula for computing the Lojasiewicz exponent of a non-constant
analytic function germ f in terms of the Newton polyhedron of f in the case where
f is non-negative and non-degenerate.
- Investigating into the sub-analytically bi-Lipschitz topological G-equivalence for
function germs from (Rn , 0) to (R, 0), where G is one of the classical Mather’s
groups.
- Giving a sufficient condition for a deformation of a polynomial function f in terms
of its Newton polyhedron at infinity to be analytically (smooth in the complex
case) trivial at infinity.
Keywords and phrases: Lojasiewicz inequalities, topological equivalences, non-degenerate
conditions, Newton polyhedra, sub-analytically, bi-Lipschitz, analytically trivial at
infinity.

ii

Acknowledgements
First and foremost, I would like to express my sincere gratitude for my supervisors,
- inh S˜i Tiˆe.p, who always support and encourage
Assoc. Prof. Pha.m Tiˆe´n So.n and Dr. D
me. Thank you for your helpful suggestions, your ideas and the interesting discussions
that we have had on the material of this work. Thank you so much.
I am also grateful to my teachers in the Department of Mathematics and Informatics
at Da Lat University. Thank for your attractive lectures which encourage me to pursue
scientific research.
Finally, I take this opportunity to express the profound gratitude from my heart to
my family.

iii


Introduction
The main purposes of this thesis are to study Lojasiewicz inequalities and topological
equivalences (local and at infinity) for a class of functions satisfying non-degenerate
conditions in terms of Newton polyhedra with some tools of Singularity Theory and
Semi-Algebraic Geometry.
In details, let f : (Rn , 0) → (R, 0) be an analytic function defined in a neighborhood
of the origin 0 ∈ Rn . The Classical Lojasiewicz inequality [2, 34] asserts that there exist
some constants δ > 0, c > 0, and l > 0 such that
|f (x)| ≥ cd(x, f −1 (0))l

x ≤ δ,

for all

where d(x, f −1 (0)) := inf{ x − y | y ∈ f −1 (0)}, and

·

denotes the usual Euclidean

norm in Rn .
The Lojasiewicz exponent of f at the origin 0 ∈ Rn , denoted by L0 (f ), is the infimum
of the exponents l satisfying the above Lojasiewicz inequality. Bochnak and Risler [8]
proved that L0 (f ) is a positive rational number. Moreover, the Lojasiewicz exponent
L0 (f ) is attained, i.e., there are some positive constants c and δ such that
|f (x)| ≥ cd(x, f −1 (0))L0 (f )

for

x ≤ δ.

The Lojasiewicz inequality plays an important role in many problems of mathematics, such as Singularity Theory (C 0 -sufficiency of jets, [7, 29, 32]); the complexity of the
representations of positive polynomials (Schm¨
udgen’s and Putinar’s Positivstellens¨atze
[41, 52]).
Hence, the computation and estimation of the Lojasiewicz exponent are interesting
problems. In the case where f is an analytic function in two variables, a formula for
computing the Lojasiewicz exponent L0 (f ) was given by Kuo in the paper [31] (see also
the paper [21] for the Lojasiewicz exponent at infinity). If f is a polynomial of degree d
iv


in n variables with an isolated zero at the origin, Gwozdziewicz [25] (see also [27]) prove
that:
L0 (f ) ≤ (d − 1)n + 1.

In the general case, when f may have a non-isolated singularity at the origin, Pha.m
[47], Kurdyka and Spodzieja [33] have the following explicit estimate:
L0 (f ) ≤ max{d(3d − 4)n−1 , 2d(3d − 3)n−2 }.
In general, as far as we know, there is no method to determine L0 (f ). In the
thesis, we will establish a formula for the Lojasiewicz exponent of non-negative and
non-degenerate analytic functions.
Another problem which attracts our studies is the classification of functions by
equivalence relations which is a fundamental problem in Singularity Theory. Many
authors have focused their attention on this problem, and many characteristics and
invariants of G-equivalence are established, where G is one of the classical Mather’s
groups [35, 36], i.e., G = A, K, C, or V.
The classification problem with respect to C 0 -G-equivalence relations has been wellstudied. For C ∞ -stable map germs, Mather in [36] proved that C 0 -K-equivalence implies
C 0 -A-equivalence. For analytic function germs with isolated singularities in two or three
variables, it is was proved by King in [26] (see also [1, 46]) that C 0 -V-equivalence implies
C 0 -A-equivalence. On the other hand, in [43], it is pointed out by Nishimura that C 0 V-equivalence of smooth functions with isolated singularities implies C 0 -K-equivalence.
Recently, new works have also treated such theme [1, 3, 4, 5, 13, 14, 15, 50, 56]. In this
work, we are interested in the sub-analytically, bi-Lipschitz C 0 -G-equivalence of continuous sub-analytic function germs from (Rn , 0) to (R, 0) and bi-Lipschitz K-equivalence
invariances of the Lojasiewicz exponent and the multiplicity. We also give a condition
for analytic function germs in terms of their Newton polyhedra to be sub-analytically
bi-Lipschitz C-equivalent to their Newton principal parts.
In the thesis, we also focus on the C 0 -sufficiency of jets. Recall that the k-jet of a
C r -function in the neighborhood of 0 ∈ Rn is identified with its k-th Taylor polynomial
at 0, then the function is called a realization of the jet. The k-jet is said to be C p sufficient in the C r class (p ≤ r), if for any two of its C r -realizations f and g there exists
a C p diffeomorphism ϕ of neighborhood of 0, such that f ◦ ϕ = g in some neighborhood
of 0. Kuiper [32] , Kuo [29], Bochnak and Lojasiewics [7] proved the followings:
v


Let f : (Rn , 0) → (R, 0) be a C k -function defined in a neighborhood of 0 ∈ Rn
with f (0) = 0. Then, two following conditions are equivalent:

(i) There are positive constants C and r such that
∇f (x) ≥ C x

k−1

for

x ≤ r.

(ii) The k-jet of f is sufficient in the C k class.
Analogous results in the case of complex analytic functions were proved by Chang
and Lu [11], Teissier [55], and by Bochnak and Kucharz [6]. Similar considerations are
also carried out for polynomial mappings in two variables in a neighborhood of infinity
by Cassou-Nogu`es and H`a [10]:
Let f be a polynomial in C[z1 , z2 ]. Then, two following conditions are equivalent:
(i) There are positive constants C and R such that
∇f (x) ≥ C x

k−1

for

x ≥ R.

such that for every polynomial P ∈

(ii) There exists a positive constant

C[z1 , z2 ] of degree less or equal k, whose modules of coefficients of monomials of degree k are less or equal , the links at infinity of almost all
fibers f −1 (λ) and (f + P )−1 (λ), λ ∈ C are isotopic.

Let us recall that the links at infinity of the fiber of f ∈ C[z1 , z2 ] is the set
f −1 (λ) ∩ {(x, y) ∈ C2 | |x|2 + |y|2 = R2 }
for R sufficiently large.
The result of Cassou-Nogu`es and H`a is recently generalised by Skalski [54], Rodak
and Spodzieja [49]. In this thesis, we will give a version at infinity of the above result
for a class of polynomial maps, which are Newton non-degenerate at infinity.
The results attended in this thesis assert that the topological, analytic and geometric
properties of analytic function germs and polynomials can be described by their Newton
polyhedra, see also [16, 17, 18, 20, 21, 24, 28, 37, 44, 45, 58, 59].
vi


In details, this thesis is divided into four chapters and a list of references.
Chapter 1 recalls some notions and results of Semi-algebraic Geometry, Sub-analytic
Geometry, Newton polyhedra and non-degeneracy conditions as well as of Differential
equations that are useful for subsequent studies.
Chapter 2 establishes a formula for computing the Lojasiewicz exponent of a nonconstant analytic function germ f in terms of its Newton polyhedron in the case where
f is non-negative and non-degenerate (see Theorem 2.3).
Chapter 3 investigates the sub-analytically bi-Lipschitz topological G-equivalence
for function germs from (Rn , 0) to (R, 0), where G is one of the classical Mather’s groups.
The mains results of this chapter are Theorem 3.3, Theorem 3.13 and Theorem 3.16.
Chapter 4 gives a sufficient condition for a deformation of a polynomial function
f in terms of its the Newton boundary at infinity to be analytically (smooth in the
complex case) trivial at infinity (see Theorem 4.2).
The results presented in this thesis are written on three papers done by myself
jointly with my supervisor, Assoc. Prof. Pha.m Tiˆe´n So.n. They were published in
the journals including International Journal of Mathematics [BP-1], Houston Journal of
Mathematics [BP-2] and Annales Polonici Mathematici [BP-3].

vii

Contents
Declaration of Authorship

i

Abstract

ii

Acknowledgements

iii

Introduction

iv

1 Preliminaries
1.1

1.2

1.3

1.4

1

Semi-algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Semi-algebraic sets and maps . . . . . . . . . . . . . . . . . . . .

1

1.1.2

The Tarski–Seidenberg theorem . . . . . . . . . . . . . . . . . . .

2

1.1.3

Cell decomposition . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.4

Other results of Semi-algebraic Geometry . . . . . . . . . . . . . .

5

Sub-analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.1

Semi-analytic sets and maps . . . . . . . . . . . . . . . . . . . . .

7

1.2.2

Sub-analytic sets and maps . . . . . . . . . . . . . . . . . . . . .

8

1.2.3

Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Newton polyhedra and non-degeneracy conditions . . . . . . . . . . . . .

10

1.3.1

Newton polyhedra and non-degeneracy conditions at the origin . .

10

1.3.2

Newton polyhedra and the Kouchnirenko non-degeneracy condition at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

viii


2 Computation of the Lojasiewicz exponent of non-negative and nondegenerate analytic functions

15

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2

Computation of the Lojasiewicz exponent of non-negative and non-degenerate

2.3

analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3 The sub-analytically topological types of function germs

29

3.1

G-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.2

Sub-analytically topological types . . . . . . . . . . . . . . . . . . . . . .

31

3.3

Bi-Lipschitz K-equivalence invariances
of the Lojasiewicz exponent and the multiplicity . . . . . . . . . . . . . .

3.4

37

Sub-analytically bi-Lipschitz C-equivalence and non-degeneracy conditions 38

4 Analytically principal part of polynomials at infinity

45

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.2

Main Theorem

46

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions

54

List of Author’s Related Papers

55

Bibliography

56

Table of Notations

61

Index

62

ix


Chapter 1
Preliminaries
This chapter recalls some notions and results of Semi-algebraic Geometry, Subanalytic Geometry, Newton polyhedra and non-degeneracy conditions, and Differential
equations. A detailed expositon, and proofs, can be found in [9, 23, 24, 28, 34, 37, 57].

1.1

Semi-algebraic Geometry

This section begins with basic definitions of semi-algebraic sets and maps. Some
notions and results of Semi-algebraic Geometry such as Tarski–Seidenberg theorem,
Cell Decomposition, Curve Selection Lemma and Lojasiewicz inequalities... are also
presented. A more detailed discussion and proofs can be found in [9, 34, 38, 40, 53, 57].

1.1.1

Semi-algebraic sets and maps

Definition 1.1. (See [9, Definition 2.1.4]) A subset of Rn is called semi-algebraic if it
is a finite union of sets of the form
{x ∈ Rn | f1 (x) = 0; fi (x) > 0,

i = 2, . . . , k},

where all fi are polynomials in R[x].
Example 1.2.

(i) The semi-algebraic subsets of R are the unions of finitely many

points and open intervals.
(ii) Any algebraic subsets of Rn (defined by polynomial equations) are semi-algebraic.
1


(iii) Let f (b, c, x) = x2 + bx + c be a polynomial. The set
{(b, c) ∈ R2 | f has exactly 2 distinct real roots}
is semi-algebraic in R2 .
(iv) The following sets are not semi-algebraic
{(x, y) ∈ R2 | y = sin x},

{(x, y) ∈ R2 | y = ex },

{(x, y) ∈ R2 | y = nx, n ∈ N}.

The following properties of semi-algebraic sets are elementary.

Proposition 1.3. Let A and B be semi-algebraic subsets of Rn . Then the sets A ∪
B, A ∩ B and Rn \ A are also semi-algebraic.
Definition 1.4. (See [9, Definition 2.2.5]) Let A ⊂ Rn and B ⊂ Rm be semi-algebraic
sets. A map f : A → B is said to be semi-algebraic if its graph
Graph(f ) = {(x, y) ∈ A × B | y = f (x)}
is semi-algebraic in Rn × Rm .

1.1.2

The Tarski–Seidenberg theorem

Theorem 1.5. (Tarski–Seidenberg theorem, [9, Theorem 2.2.1]) Let A be a semialgebraic subset of Rn+m and π : Rn ×Rm → Rn , the projection on the first n coordinates.
Then π(A) is a semi-algebraic subset of Rn .
Let x, y, z be variables ranging over the sets X, Y, Z, respectively and let φ(x, y, z)
and ϕ(x, y, z) be formulas (conditions on (x, y, z)) defining the sets
φ := {(x, y, z) ∈ X × Y × Z | φ(x, y, z) holds },
ϕ := {(x, y, z) ∈ X × Y × Z | ϕ(x, y, z) holds }.
Then we can construct new formulas as below:
• The disjunction of φ and ϕ, denoted by φ ∨ ϕ, defines the set φ ∪ ϕ.
• The conjunction of φ and ϕ, denoted by φ ∧ ϕ, defines the set φ ∩ ϕ.
2


• The negation of φ, denoted by ¬φ, defines the complement X × Y × Z \ φ.
• The existential quantification over z of φ(x, y, z), denoted by ∃zφ(x, y, z), defines
the set {(x, y) ∈ X × Y | there exists z ∈ Z such that φ(x, y, z) holds }.
• The universal quantification over z of φ(x, y, z), denoted by ∀zφ(x, y, z), defines
the set {(x, y) ∈ X × Y | for all z ∈ Z the condition φ(x, y, z) holds }.
Definition 1.6. A first-order formula (of the language of ordered fields with parameters
in R) is obtained by the following rules.

(1) If f ∈ R[x1 , . . . , xn ], then f = 0 and f > 0 are first-order formulas.
(2) If φ and ϕ are first-order formulas, then φ ∨ ϕ, φ ∧ ϕ and ¬φ are also first-order
formulas.
(3) If φ is a first-order formula and x is a variable ranging over R, then ∃xφ and ∀xφ
are first-order formulas.
The formulas obtained by using only rules (1) and (2) are called quantifier-free formulas.
With the above notion, we have
Theorem 1.7 (Logical formulation of the Tarski–Seidenberg theorem). If φ(x) is a
first-order formula, then the set {x ∈ Rn | φ(x) holds} is semi-algebraic.
The following properties of semi-algebraic sets and maps follow from the Tarski–
Seidenberg theorem.
Proposition 1.8. ([9, Proposition 2.2.2]) The following two statements hold.
(i) If A and B are semi-algebraic sets, then A × B is also semi-algebraic.
(ii) The closure, the interior and the boundary of a semi-algebraic set are semi-algebraic.
(iii) Images and inverse images of semi-algebraic sets under semi-algebraic maps are
semi-algebraic.
(iv) Compositions of semi-algebraic maps are semi-algebraic.
(v) The sum and product of two semi-algebraic functions are semi-algebraic.
3


Example 1.9.

(i) Let A ⊂ Rn be a semi-algebraic set. If F : A → Rm is a polynomial

mapping, it is semi-algebraic.
(ii) Let A ⊂ Rn , A = ∅ be a semi-algebraic set. Then the distance function
d(·, A) : Rn → R,

x → d(x, A) := inf{ x − a | a ∈ A}

is continuous semi-algebraic.

1.1.3

Cell decomposition

Analytic cells are non-empty semi-algebraic sets of an especially simple nature. They
are defined inductively as follows:
• Analytic cells in R are points or open intervals (a, b), −∞ ≤ a < b ≤ +∞.
• Let C ⊂ Rn−1 be an analytic cell. If f, g : C → R are analytic semi-algebraic
functions such that f < g on C, then the cylinder
(f, g) := {(x, t) ∈ C × R | f (x) < t < g(x)}
as well as
(−∞, f ) := {(x, t) ∈ C × R | − ∞ < t < f (x)}
and
(g, +∞) := {(x, t) ∈ C × R | g(x) < t < +∞}
are analytic cells in Rn . If f : C → R is an analytic semi-algebraic function, then
its graph
Graph(f ) = {(x, t) ∈ C × R | t = f (x)}
is an analytic cells in Rn . Finally, C × R ⊂ Rn is an analytic cell.
An analytic cell decomposition of Rn is defined by induction on n:
• An analytic cell decomposition of R is a finite collection of open intervals and
points:
{(−∞, a1 ), {a1 }, (a1 , a2 ), . . . , {ak }, (ak , +∞)},
where a1 < a2 < · · · < ak are points in R.
4


• Assuming that the class of analytic cell decomposition of Rn−1 has been defined,

an analytic cell decomposition of Rn is finite partition P of Rn into analytic cells
such that
π(P) := {π(C) | C ∈ P}
is an analytic cell decomposition of Rn−1 , where π : Rn → Rn−1 is the projection
on the first (n − 1) coordinates.
We say that a decomposition P of Rn partitions a set A ⊂ Rn if A is a disjoint union of
cells in P.
Theorem 1.10. (Analytic Cell Decomposition, [57, Theorem 2.11]) Let A1 , . . . , Ak be
semi-algebraic subsets of Rn . Then there is an analytic cell decomposition of Rn partitioning each Ai .
Theorem 1.11. (Finiteness of the number of connected components, [57, Proposition
2.18]) Every semi-algebraic set has a finite number of connected components and each
such component is semi-algebraic.

1.1.4

Other results of Semi-algebraic Geometry

Theorem 1.12. (Monotonicity Theorem, [57, Theorem 1.3]) Let f : (a, b) → R be a
semi-algebraic function. Then there are a = a0 < a1 < · · · < as < as+1 = b such that,
for each i = 0, . . . , s, the restriction f|(ai ,ai+1 ) is analytic, and either constant, or strictly
monotone.
Theorem 1.13. Curve Selection Lemma (Monotonicity Theorem, [9, Theorem 2.5.5])
Let S be a semi-algebraic subset of Rn and x0 ∈ S¯ \ S. Then there exists a real analytic
semi-algebraic curve
φ : (0, ) → S
with φ(0) = x0 and with φ(t) ∈ S for t ∈ (0, ).
Theorem 1.14 (Curve Selection Lemma at infinity, [40]). Let S ⊂ Rn be a semialgebraic set, and let
f := (f1 , . . . , fp ) : Rn → Rp

5

be a semi-algebraic map. Assume that there exists a sequence {xk } such that xk ∈
S, limk→∞ xk = ∞ and limk→∞ f (xk ) = y ∈ (R)p , where R := R ∪ {±∞}. Then there
exists a smooth semi-algebraic curve
φ : (0, ) → Rn
such that φ(t) ∈ S for all t ∈ (0, ), limt→0 γ(t) = ∞ and limt→0 f (γ(t)) = y.
(i) Let f : (0, ) → R be a semi-

Theorem 1.15 (Growth Dichotomy Lemma, [40]).

algebraic function with f (t) = 0 for all t ∈ (0, ). Then there exists some constants
a = 0 and α ∈ Q such that f (t) = atα + o(tα ) as t → 0+ .
(ii) Let f : (r, +∞) → R be a semi-algebraic function with f (t) = 0 for all t ∈ (r, +∞).
Then there exist some constants a = 0 and α ∈ Q such that f (t) = atα + o(tα ) as
t → +∞.
Theorem 1.16. (Lojasiewicz inequality, [9, Corollary 2.6.7]) Let K be a compact semialgebraic subset of Rn . Let f, g : K → R be continuous semi-algebraic functions such
that f −1 (0) ⊂ g −1 (0). Then there exist α > 0 and C > 0 such that
|f (x)| ≥ C|g(x)|α

∀x ∈ K.

Theorem 1.17 (Classical Lojasiewicz inequality). Let K be a compact semi-algebraic
subset of Rn and f : K → R be a continuous semi-algebraic function. Then there exist
α > 0 and C > 0 such that
|f (x)| ≥ Cd(x, f −1 (0))α

∀x ∈ K.

Theorem 1.18 (Lojasiewicz gradient inequality). Let f be a semi-algebraic function

of class C 1 in a neighborhood of 0 ∈ Rn such that f (0) = 0. Then there exist some
constants C > 0 and α ∈ [0, 1) such that, for all x in a neighborhood of 0,
|∇f (x)| ≥ C|f (x)|α .

1.2

Sub-analytic Geometry

We begin by recalling some notions of semi-analytic sets and maps. Then, after
presenting some definitions and properties of sub-analytic sets and maps, we complete
this section by the results of triangulation. For more details the reader is referred to
[2, 53, 57].
6


1.2.1

Semi-analytic sets and maps

Let M be a real analytic manifold, Hausdorff with a countable basis. The ring of
real analytic functions on M will be denoted by O(M ).
Definition 1.19. ([2, Definition 2.1]) A subset X of M is called semi-analytic if each
a ∈ M has a neighborhood U such that
p

q

{x ∈ U : fij (x) sij 0},

X ∩U =

i=1 j=1

where p, q ∈ N, fij ∈ O(U ) and sij ∈ {=, >}, 1

i

p, 1

j

q.

Definition 1.20. ([2, Definition 2.3]) Let X be a subset of M . A map f : X → Rm is
said to be semi-analytic if its graph
Graph(f ) = {(x, y) ∈ X × Rm | y = f (x)}
is a semi-analytic subset of M × Rm .
Example 1.21.

(i) An analytic subset of M (defined by analytic equations) is semi-

analytic in M .
(ii) A semi-algebraic subset of Rn is semi-analytic in Rn .
(iii) The set
{(x, y) ∈ R2 | y = sin x}
is semi-analytic but not semi-algebraic in R2 .
(iv) The set
1

{(x, e− x ) | x ∈ R, x > 0}

is semi-analytic in R2 \ {(0, 0)} but not semi-analytic in R2 .
Proposition 1.22. ([2, Corollary 2.7]) Let X be a semi-analytic subset of M . Then
(i) The closure, the interior and the boundary of X are semi-analytic.
(ii) Every connected component of X is semi-analytic.
(iii) The family of connected components of X is locally finite (in particular, finite if
X is relatively compact).
7


(iv) X is locally connected.
Remark 1.23. It was shown by the example of Osgood ([2, Example 2.14]) that there
exists a semi-analytic subset of R5 whose image by the projection map is not semianalytic. Thus, the Tarski–Seidenberg theorem is false in genaral for semi-analytic sets.

1.2.2

Sub-analytic sets and maps

Let M be a real analytic manifold, Hausdorff with a countable basis.
Definition 1.24. (See [2, Definition 3.1]) A subset X of M is called sub-analytic if each
a ∈ M has a neighborhood U such that X ∩ U is a projection of a relatively compact
semi-analytic set in M × Rk (with k depend on a).
Proposition 1.25. The following statements hold.
(i) Let X and Y be sub-analytic subsets of M . Then the sets X ∪ Y, X ∩ Y are also
sub-analytic.
(ii) Let X be a sub-analytic subset of M and let N be a real analytic manifold. If Z is
a sub-analytic subset of N , then X × Z is sub-analytic in M × N .
Proposition 1.26. Let X be a sub-analytic subset of M . Then
(i) The closure of X is sub-analytic.
(ii) Every connected component of X is sub-analytic.
(iii) The family of connected components of X is locally finite.

(iv) X is locally connected.
Definition 1.27. ([2, Definition 3.2]) Let X be a sub-analytic subset of M and N be a
real analytic manifold. A map f : X → N is called sub-analytic if its graph
Graph(f ) = {(x, y) ∈ X × N | y = f (x)}
is sub-analytic in M × N .
Proposition 1.28. Let f : X → N be a sub-analytic map. Then:
8


(i) If A be a relatively compact sub-analytic subset of M , then f (A) is sub-analytic.
(ii) If B be a relatively compact sub-analytic subset of N , f −1 (B) is sub-analytic.
Theorem 1.29. (Theorem of the complement, [2, Theorem 3.10]) Let X be a subanalytic subset of M . Then M \ X is sub-analytic.
Remark 1.30. The versions of the Cell decomposition, Cure selection Lemma and
Lojasiewicz inequalities still hold in the sub-analytic context which can be found in [2].

1.2.3

Triangulation

We first recall some definitions concerning simplicial complexes. Let a0 , . . . , ak be
k + 1 points of Rn which are affine independent (i.e. the affine span of a0 , . . . , ak has
dimension k). The k−simplex with vertices a0 , . . . , ak is
k

k

n

(a0 , . . . , ak ) := {x ∈ R | x =

t i ai ,

t0 , . . . , tk > 0,

i=0

ti = 1}.
i=0

The closure of (a0 , . . . , ak ) in Rn is denoted by [a0 , . . . , ak ], so that
k
n

[a0 , . . . , ak ] := {x ∈ R | x =

k

ti ai ,
i=0

t0 , . . . , tk ≥ 0,

ti = 1}.
i=0

We call a0 , . . . , ak the vertices of (a0 , . . . , ak ) and also of [a0 , . . . , ak ]. A face of the simplex
σ = (a0 , . . . , ak ) is a simplex spanned by a non-empty subset of {a0 , . . . , ak }.
A finite simplicial complex in Rn is a finite K = {σ1 , . . . , σp } of simplices σi ⊂ Rn such
that, for every σi , σj ∈ K, either the intersection σ¯i ∩ σ¯j is a common face of σ¯i and σ¯j ,
or σ¯i ∩ σ¯j = ∅.

We set |K| := ∪σi ∈K σi ; which is a semi-algebraic subset of Rn . A polyhedron in Rn is a
subset P of Rn , such that there exists a finite simplicial complex K in Rn with P = |K|.
For a sub-analytic set X ⊂ Rm , a sub-analytic triangulation of X is a pair (K, h),
where K is a simplicial complex with |K| closed in Rn , and h : |K| → X is a sub-analytic
homeomorphism such that for each simplex σ in K, h|σ is a C r -diffeomorphism onto the
image. A sub-analytic triangulation of a a sub-analytic function on X is a sub-analytic
triangulation (K, h) of X such that for each simplex σ in K, f ◦ h|σ is linear.

9


Theorem 1.31 (Triangulation of sub-analytic sets, [22]). Let X1 , . . . , Xk be bounded
sub-analytic subsets of Rm . Then there exists a simplicial complex K in Rn and a subanalytic homeomorphism h : |K| → X, such that for each simplex σ in K, h|σ : σ → h(σ)
is an analytic diffeomorphism and each Xi is the image by h of a union simplices of K.
Theorem 1.32. (Triangulation of sub-analytic functions, [53, Theorem II.3.1]) Let X
be a closed sub-analytic subset of Rm and f : X → R be a sub-analytic function. Then
there exists a sub-analytic triangulation of f .

1.3

Newton polyhedra and non-degeneracy conditions

This section covers topics: Newton polyhedra and non-degeneracy conditions (at
the origin and at infinity).
Throughout this section, we consider a fixed coordinate system x1 , . . . , xn ∈ Rn .
We denote by R+ the set of non-negative real numbers and Z+ := Z ∩ R+ the set of
non-negative integer numbers. If x = (x1 , . . . , xn ) ∈ Rn and α = (α1 , . . . , αn ) ∈ Zn+ , we
denote the monomial xα1 1 . . . xαnn by xα and by |α| the sum α1 + · · · + αn .

1.3.1

Newton polyhedra and non-degeneracy conditions at the
origin

Let f : (Rn , 0) → (R, 0) be an analytic function germ. We express f as follows:
f (x) :=

α cα x

α

for some cα ∈ R. The support supp(f ) is defined to be {α ∈ Zn+ : cα =

0}. We denote Γ+ (f ), which be called the Newton polyhedron of f at the origin, to be
the convex hull of the set {α + Rn+ : α ∈ supp(f )}. If f ≡ 0, then we set Γ+ (f ) = ∅.
We say that f is convenient if Γ+ (f ) intersects each coordinate axis at a point
different from the origin, i.e., if for any i ∈ {1, . . . , n} there exists some integer mj > 0
such that mj ej ∈ Γ+ (f ).
Given an analytic function germ f : (Rn , 0) → (R, 0). Let Γ+ (f ) be the Newton

10


polyhedron of f and a vector a ∈ Rn , a = 0, we define
d(a, Γ+ (f )) := min{ a, α | α ∈ Γ+ (f )},
∆(a, Γ+ (f )) := {α ∈ Γ+ (f ) | a, α = d(a, Γ+ (f ))}.
We say that a subset ∆ of Γ+ (f ) is a face of Γ+ (f ) if there exists a vector a ∈ Rn , a = 0,
such that ∆ = ∆(a, Γ+ (f )). The dimension of a face ∆ is defined as the minimum of the
dimensions of the affine subspaces containing ∆. The faces of dimension 0 of Γ+ (f ) are
called the vertices of Γ+ (f ).

The Newton boundary of f at the origin, denoted by Γ(f ), is by definition the
collection of all compact faces of Γ+ (f ). For each face ∆ ∈ Γ(f ), we denote by f∆ (x)
the polynomial

α∈∆ cα x

α

. In particular, fΓ (x) :=

α∈Γ(f ) cα x

α

.

Remark 1.33. By definition, the number of faces of the Newton boundary Γ(f ) is finite.
We now introduce two different notions of non-degeneracy.
Definition 1.34 (see [24], [37]). We say that f is Mikhailov–Gindikin non-degenerate
((M-G) for short) if and only if for any face ∆ ∈ Γ(f ) we have
f∆ (x) = 0

for all

x ∈ (R \ {0})n .

Definition 1.35 (see [28]). We say that f is Kouchnirenko non-degenerate ((K) for
short) if and only if for any face ∆ ∈ Γ(f ), the system of equations
∂f∆ (x)
∂f∆ (x)

= ··· =
=0
∂x1
∂xn
has no solutions in (R \ {0})n .
Remark 1.36. It is well known (see [28]) that the class of analytic functions, which
is Kouchnirenko non-degenerate, is open and dense (in fact, Zariski-open) when the
Newton boundary is fixed.

1.3.2

Newton polyhedra and the Kouchnirenko non-degeneracy
condition at infinity

Let f : Kn → K be a polynomial function, where K := R or C. Suppose that f is
written as f =

α

aα xα . Then the support of f, denoted by supp(f ), is defined as the
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set of those α ∈ Zn+ such that aα = 0. The Newton polyhedron of f at infinity, denoted
by Γ− (f ), is the convex hull of the set supp(f ). If f ≡ 0, then we set Γ− (f ) = ∅.
We say that f is convenient if Γ− (f ) intersects each coordinate axis in a point
different from the origin, i.e., if for any i ∈ {1, . . . , n} there exists some integer mj > 0
such that mj ej ∈ Γ− (f ).
Given a polynomial function f : Kn → K. Let Γ− (f ) be the Newton polyhedron at
infinity of f and a vector a ∈ Rn , a = 0, we define

d(a, Γ− (f )) := min{ a, α | α ∈ Γ− (f )},
∆(a, Γ− (f )) := {α ∈ Γ− (f ) | a, α = d(a, Γ− (f ))}.
We say that a subset ∆ of Γ− (f ) is a face of Γ− (f ) if there exists a vector a ∈ Rn , a = 0,
such that ∆ = ∆(a, Γ− (f )). The dimension of a face ∆ is defined as the minimum of the
dimensions of the affine subspaces containing ∆. The faces of Γ− (f ) of dimension 0 are
called the vertices of Γ− (f ).
The Newton boundary of f at infinity, denoted by Γ∞ (f ), is defined as the union of
the closed faces of Γ− (f ) which do not contain the origin in Rn . For each closed face ∆
of Γ∞ (f ) we denote by f∆ the polynomial

α∈∆

aα xα .

Definition 1.37 ( [28]). We say that f is (Kouchnirenko) non-degenerate at infinity if
and only if for any face ∆ ∈ Γ∞ (f ), the system of equations
∂f∆ (x)
∂f∆ (x)
= ··· =
=0
∂x1
∂xn
has no solutions in (R \ {0})n .
Remark 1.38. The condition above is generic in the sense that the class of polynomial
maps (with fix Newton polyhedra), which is non-degenerate at infinity, is an open and
dense semi-algebraic set ([28, 45]).

1.4

Differential equations

The aim of this section is to present some results concerning the existence, uniqueness, continuity of solutions for differential equations, which can be found in [23].

12


Throughout this section, E will denote a normed vector space, U is an open set
in R × E and F : U → E a continuous map. Let (t0 , x0 ) ∈ U . By a solution of the
differential equation with the initial condition
dx
(t) = F (t, x(t)),
dt

x(t0 ) = x0 ,

(1.1)

we mean a differentiable map
x : J → U,
defined on some interval J ⊂ R such that
t0 ∈ J
(t, x(t)) ∈ U,

and x(t0 ) = x0 ,
dx
(t) = F (t, x(t)),
dt

for all t ∈ J.
Definition 1.39. A map F : U → E is called Lipschitz in x if there is a constant L ≥ 0

such that
F (t, x1 ) − F (t, x2 ) ≤ L x1 − x2 , ∀ (t, x1 ), (t, x2 ) ∈ U.
More generally, the map F is called locally Lipschitz in x if each point of U has a
neighborhood U0 in U such that the restriction F |U0 is Lipschitz in x.
Theorem 1.40. Let F : U → E be a continuous map which is locally Lipschitz in x.
Let (t0 , x0 ) ∈ U , then there exist an open interval J containing t0 and a unique solution
to (1.1) defined on J.
Another problem we shall consider is the continuation of the solution x(t). If there
exists another solution v(t) of (1.1) such that the domain of v(t) contains the interval
(t1 , t2 ) and x(t) = v(t) for all t ∈ (t1 , t2 ), then x(t) is called a continuable solution.
The solution v(t) is called a continuation of the solution x(t). If the solution x(t) is not
continuable, then it is called maximal.
Theorem 1.41. Suppose that two solutions x(t), v(t) of (1.1) are defined on the same
open interval J ∈ R containing t0 and satisfy x(t0 ) = v(t0 ). Then x(t) = v(t) for all
t ∈ J.
Theorem 1.42. Let x(t) be a continuable solution of (1.1). Then there exists a unique
maximal solution of (1.1), continuing x(t).
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Finally, we consider the dependence of solutions on initial values. Let F : U → E
be a continuous map. Assume that F is locally Lipschitz in x. Let (t0 , x0 ) ∈ U , then
there exists a unique maximal solution passing through this point. Let us denoted it
by x = x(t, t0 , x0 ), to make clear the fact that it depends on the initial condition. As a
function of t, x(t, t0 , x0 ) is defined on some open interval J containing t0 . Thus, for all
t ∈ J, we have:
dx
(t, t0 , x0 ) = F (t, x(t, t0 , x0 )) and x(t0 , t0 , x0 ) = x0 .
dt
If ξ = x(τ, t0 , x0 ) then

x(t, τ, ξ) = x(t, t0 , x0 ).

Let
Ω := {(τ, η, t) ∈ U × R | (τ, η) ∈ U, t ∈ J(τ, η)}.
The map
Φ : Ω → U, (τ, η, t) → x(τ, η)(t)
is called the global solution of (1.1).
Theorem 1.43. If the map F is C m (m ≥ 1), then there exists a C m global solution of
(1.1).
The following result can be founded in [39, Theorem 1.8.12].
Theorem 1.44. If the map F is analytic, then there exists an analytic global solution
of (1.1).

14