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: 62.02 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: 62.02 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisors: 1. Pham Tien Son 2. Dinh Si Tiep DALAT, 2017 Declaration of Authorship I, Bùi Nguyê˜n Thȧ’o Nguyên, declare that this thesis titled, “LOJASIEWICZ IN- EQUALITIES, 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 at- tributed. • 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 Bùi Nguyê˜n Thȧ’o Nguyên The goals of this thesis are to study Lojasiewicz inequalities and topological equiva- lences (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, Assoc. D - inh Sĩ Tiê.p, who always support and encourage 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 for all kxk ≤ δ, where d(x, f −1 (0)) := inf{kx − yk | y ∈ f −1 (0)}, and k · k 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., there are some positive constants c and δ such that |f (x)| ≥ cd(x, f −1 (0))L0 (f ) for kxk ≤ δ. The Lojasiewicz inequality plays an important role in many problems of mathemat- ics, such as Singularity Theory (C 0 -sufficiency of jets, [7, 29, 32]); the complexity of the representations of positive polynomials (Schmüdgen’s and Putinar’s Positivstellensätze [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. 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. The classification problem with respect to C 0 -G-equivalence relations has been well- studied. 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 contin- uous 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.

Then, two following conditions are equivalent: (i) There are positive constants C and r such that k∇f (x)k ≥ Ckxkk−1 for kxk ≤ 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-Noguès and Hà [10]: Let f be a polynomial in C[z1 , z2 ].

Then, two following conditions are equiv- alent: (i) There are positive constants C and R such that k∇f (x)k ≥ Ckxkk−1 for kxk ≥ R. 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-Noguès and Hà 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 non- constant analytic function germ f in terms of its Newton polyhedron in the case where f is non-negative and non-degenerate (see Theorem 2.

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. 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. The results presented in this thesis are written on three papers done by myself jointly with my supervisor, Assoc.

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 Semi-algebraic Geometry .1 Semi-algebraic sets and maps .2 The Tarski–Seidenberg theorem .4 Other results of Semi-algebraic Geometry .2 Sub-analytic Geometry .1 Semi-analytic sets and maps .2 Sub-analytic sets and maps .3 Newton polyhedra and non-degeneracy conditions .1 Newton polyhedra and non-degeneracy conditions at the origin .2 Newton polyhedra and the Kouchnirenko non-degeneracy condi- tion at infinity. 12 viii 2 Computation of the Lojasiewicz exponent of non-negative and non- degenerate analytic functions 15 2.2 Computation of the Lojasiewicz exponent of non-negative and non-degenerate analytic functions. 27 3 The sub-analytically topological types of function germs 29 3.2 Sub-analytically topological types .3 Bi-Lipschitz K-equivalence invariances of the Lojasiewicz exponent and the multiplicity .4 Sub-analytically bi-Lipschitz C-equivalence and non-degeneracy conditions 38 4 Analytically principal part of polynomials at infinity 45 4.

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, Sub- analytic 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 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 Semi-algebraic sets and maps Definition 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]. (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.