Lojasiewicz inequalities, topological equivalences and newton polyhedra

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: Assoc Prof Pham Tien Son Dr Dinh Si Tiep DALAT, 2017 Declaration of Authorship I, B` ui Nguyˆ˜en Tha˙’o Nguyên, 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 Nguyên 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 Tiê.p, who always support and encourage Assoc Prof Pha.m Tiêń 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 C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an 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 ∈ Rn The Classical Lojasiewicz inequality [2, 34] asserts that there exist some constants δ > 0, c > 0, and l > 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 ∈ 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 kxk ≤ δ The Lojasiewicz inequality plays an important role in many problems of mathematics, such as Singularity Theory (C -sufficiency of jets, [7, 29, 32]); the complexity of the representations of positive polynomials (Schmă udgens and Putinars Positivstellensăatze [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 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an in n variables with an isolated zero at the origin, Gwozdziewicz [25] (see also [27]) prove that: L0 (f ) ≤ (d − 1)n + 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 -G-equivalence relations has been wellstudied For C ∞ -stable map germs, Mather in [36] proved that C -K-equivalence implies C -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 -V-equivalence implies C -A-equivalence On the other hand, in [43], it is pointed out by Nishimura that C V-equivalence of smooth functions with isolated singularities implies C -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 -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 -sufficiency of jets Recall that the k-jet of a C r -function in the neighborhood of ∈ 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 Kuiper [32] , Kuo [29], Bochnak and Lojasiewics [7] proved the followings: v Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Let f : (Rn , 0) → (R, 0) be a C k -function defined in a neighborhood of ∈ Rn with f (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 equivalent: (i) There are positive constants C and R such that k∇f (x)k ≥ Ckxkk−1 for kxk ≥ R (ii) There exists a positive constant such that for every polynomial P ∈ 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-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 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an In details, this thesis is divided into four chapters and a list of references Chapter 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 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 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 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 Tiêń 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 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Contents Declaration of Authorship i Abstract ii Acknowledgements iii Introduction iv Preliminaries 1.1 1.2 1.3 1.4 Semi-algebraic Geometry 1.1.1 Semi-algebraic sets and maps 1.1.2 The Tarski–Seidenberg theorem 1.1.3 Cell decomposition 1.1.4 Other results of Semi-algebraic Geometry Sub-analytic Geometry 1.2.1 Semi-analytic sets and maps 1.2.2 Sub-analytic sets and maps 1.2.3 Triangulation 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 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Proof By contradiction and using the Curve Selection Lemma at infinity (Theorem 1.14), we can find an analytic function t(s) and an analytic curve ϕ(s) = (ϕ1 (s), , ϕn (s)) for s ∈ (0, ) satisfying the following conditions: (a) −2 < t(s) < 2; (b) kϕ(s)k → ∞ as s → 0+ ; and (c) kGradx F (t(s), ϕ(s))k2 = Pn i=1 2 ∂F ϕi (s) ∂x (t(s), ϕ(s)) |g(ϕ(s))|2 i Let I := {i | ϕi 6≡ 0} ⊂ {1, , n} By Condition (b), J 6= ∅ For i ∈ I, we can expand the coordinate ϕi in terms of the parameter: say ϕi (s) = x0i sai + higher order terms in s, where x0i 6= and ∈ Q From Condition (b), we obtain mini∈I < Let RI := {α = (α1 , , αn ) ∈ Rn | αi = for i 6∈ I} Since f is convenient, Γ− (f )∩RI 6= ∅ We put a := (a1 , , an ) ∈ Rn , where := for i 6∈ I Let d be the minimal value of P the linear function ha, αi = i∈I αi on Γ− (f ) ∩ RI , and let ∆ be the (unique) maximal P face of Γ− (f ) where the linear function i∈I αi takes its minimal value d, i.e., ∆ := {α ∈ Γ− (f ) | ha, αi = d} We have d < and ∆ is a closed face of Γ∞ (f ) because f is convenient and mini=1, ,n < Since Γ− (g) ⊂ Int(Γ− (f )) and t(s) is bounded, we can see that ϕi (s) ∂F ∂f∆ d (t(s), ϕ(s)) = x0i (x )s + higher order terms in s, ∂xi ∂xi for i ∈ I, where x0 := (x01 , , x0n ) and x0i := for i 6∈ I Consequently, we obtain 2 n X ∂F kGradx F (t(s), ϕ(s))k = ϕi (s) (t(s), ϕ(s)) ∂x i i=1 2 X ∂F = ϕi (s) (t(s), ϕ(s)) ∂xi i∈I 2 X ∂f∆ 2d 0 = s xi (x ) + higher order terms in s, ∂xi i∈I 2 n X ∂f∆ 2d xi = s (x ) + higher order terms in s ∂xi i=1 48 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an (The last equality follows from the fact that f∆ does not depend on xi for i ∈ / I.) Since the polynomial f is non-degenerate at infinity, it holds that 2 n X 0 ∂f∆ (x ) 6= xi ∂xi i=1 Therefore kGradx F (t(s), ϕ(s))k ' sd as s → 0+ (4.1) On the other hand, it follows easily from the assumption Γ− (g) ⊂ Int(Γ− (f )) that g(ϕ(s)) = o(sd ) as s → 0+ This, together with (4.1), contradicts Condition (c) Lemma 4.4 There exist positive constants C and R such that kGradx F (t, x)k ≥ C, ∀ t ∈ (−2, 2), ∀ kxk > R Proof By contradiction and using the Curve Selection Lemma at infinity, we can find an analytic function t(s) and an analytic curve ϕ(s) = (ϕ1 (s), , ϕn (s)) for s ∈ (0, ) satisfying the following conditions (a) −2 < t(s) < 2; (b) kϕ(s)k → ∞ as s → 0+ ; and (c) kGradx F (t(s), ϕ(s))k2 = Pn i=1 2 ∂F ϕi (s) ∂x (t(s), ϕ(s)) → as s → 0+ i Let I := {i | ϕi 6≡ 0} ⊂ {1, , n} By Condition (b), I 6= ∅ For i ∈ I, we can expand the coordinate ϕi in terms of the parameter: say ϕi (s) = x0i sai + higher order terms in s, where x0i 6= and ∈ Q From Condition (b), we obtain mini∈I < Let RI := {α = (α1 , , αn ) ∈ Rn | αi = for i 6∈ I} Since the polynomial f is convenient, we have Γ− (f ) ∩ RI 6= ∅ We put a := (a1 , , an ) ∈ Rn , where = for P i 6∈ I Let d be the minimal value of the linear function i∈I αi on Γ− (f ) ∩ RI , and P let ∆ be the (unique) maximal face of Γ− (f ) where the linear function i∈I αi takes 49 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an its minimal value d Then it is easy to see that d < and ∆ is a closed face of Γ∞ (f ) Since Γ− (g) ⊂ Int(Γ− (f )) and t(s) is bounded, we can see that ϕi (s) ∂f∆ d ∂F (t(s), ϕ(s)) = x0i (x )s + higher order terms in s, ∂xi ∂xi for i ∈ I, where x0 := (x01 , , x0n ) and x0i := for i 6∈ I Then it follows from Condition (c) that x0i ∂f∆ (x ) = 0, ∂xi i for i ∈ I Note that f∆ does not depend on xi for i 6∈ I Therefore, x0i ∂f∆ (x ) = 0, ∂xi i for i = 1, , n This contradicts our assumption that f is non-degenerate at infinity Lemma 4.5 For each index i = 1, , n, let  −g(x)  x2 ∂F (t, x) kGradx F (t,x)k2 i ∂xi Wi (t, x) := 0 if Gradx F (t, x) 6= 0, otherwise Then there exist positive constants C and R such that the mapping W : (−2, 2) × {x ∈ Rn | kxk > R} → Rn , (t, x) 7→ (W1 (t, x), , Wn (t, x)), is analytic and satisfies the following conditions kW (t, x)k ≤ Ckxk, dx F (t, x)W (t, x) = −g(x), where dx F stands for the derivative of F with respect to the variable x Proof Let C and R be positive constants for which Lemmas 4.3 and 4.4 hold true Then the mapping W is well-defined and analytic Furthermore, we have for all t ∈ (−2, 2) and all kxk > R, n X ∂F |g(x)| kW (t, x)k ≤ |xi | xi (t, x) kGradx F (t, x)k i=1 ∂xi v v u n u n 2 uX uX |g(x)| ∂F 2t t ≤ x xi (t, x) kGradx F (t, x)k2 i=1 i i=1 ∂xi = |g(x)| kxk kGradx F (t, x)k 50 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an By Lemma 4.3, we conclude that kW (t, x)k ≤ kxk C Finally, we have for t ∈ (−2, 2) and kxk > R, n X ∂F dx F (t, x)W (t, x) = (t, x)Wi (t, x) ∂xi i=1 = n X i=1 −g(x) 2 Pn ∂F x (t, x) k ∂xk k=1 2 ∂F xi (t, x) ∂xi = −g(x) This completes the proof of the lemma Lemma 4.6 (See [54, Lemmas and 2]) Let D := {x ∈ Rn | kxk > R}, R > 0, and let W : (−2, 2) × D → Rn be a continuous mapping such that for some C > we have kW (t, x)k ≤ Ckxk for all (t, x) ∈ (−2, 2) × D Assume that h : (α, β) → D is a maximal solution of the system of differential equations y (t) = W (t, y(t)) If ∈ (α, β), h(0) = x, and kxk > ReC , then β > and e−C kxk ≤ kh(1)k ≤ eC kxk If ∈ (α, β), h(1) = x, and kxk > ReC , then α < and e−C kxk ≤ kh(0)k ≤ eC kxk Now, we are in a position to finish the proof of Theorem 4.2 Proof of Theorem 4.2 We first consider the real case There exist positive constants C and R such that the mapping W : (−2, 2) × {x ∈ Rn | kxk > R} → Rn defined in Lemma 4.5 is analytic Let us consider the following system of differential equations y (t) = W (t, y(t)) 51 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn (4.2) C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Let D := {x ∈ Rn | kxk > R} Since the mapping W is analytic in (−2, 2) × D, it follows from Theorem 1.44 that for any (s, x) ∈ (−2, 2) × D, there exists a unique solution φ(s,x) of (4.2) defined on an open interval I(s, x) of R and satisfying the initial condition φ(s,x) (s) = x Moreover, the mapping {(s, x, t) ∈ R × Rn × R | s ∈ (−2, 2), kxk > R, t ∈ I(s, x)} → Rn (s, x, t) 7→ φ(s,x) (t) is analytic Set Ω := {x ∈ Rn | kxk > ReC } and define the mappings Φ, Ψ : [0, 1] × Ω → D by Φ(t, x) = φ(0,x) (t) and Ψ(t, y) = φ(t,y) (0) By Lemmas 4.5 and 4.6, the mappings Φ, Ψ are well defined, analytic, and satisfy e−C kxk ≤ kΦ(t, x)k ≤ eC kxk, for t ∈ [0, 1], x ∈ Ω, e−C kyk ≤ kΨ(t, y)k ≤ eC kyk, for t ∈ [0, 1], y ∈ Ω (4.3) Let Ω0 := {x ∈ Rn | kxk > Re2C } and Ωt := {y ∈ Rn | Ψ(t, y) ∈ Ω0 } for t ∈ (0, 1] Then for each t ∈ [0, 1], Ωt is an open subset of Ω and the following inclusion holds true {y ∈ Rn | kyk > Re3C } ⊂ Ωt , and therefore Ωt is a neighborhood of infinity On the other hand, from the global uniqueness of solutions of (4.2), it is easy to see that for any t ∈ [0, 1] we have (a) Φ(0, x) = x for x ∈ Ω0 ; (b) Ψ(t, Φ(t, x)) = x for x ∈ Ω0 ; and (c) Φ(t, Ψ(t, y)) = y for y ∈ Ωt Summing up, for each t ∈ [0, 1] the mappings Φt : Ω0 → Ωt , x 7→ Φ(t, x), and Ψt : Ωt → Ω0 , y 7→ Ψ(t, y), are analytic diffeomorphisms of neighborhoods of infinity and Ψt = Φ−1 t Moreover, by Inequality (4.3), we have kΦt (x)k → ∞ if, and only if, kxk → ∞ 52 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Finally, thanks to Lemma 4.5, we obtain dF (t, φ(0,x) (t)) = g(φ(0,x) (t)) + dx F (t, φ(0,x) (t))φ0(0,x) (t) dt = g(φ(0,x) (t)) + dx F (t, φ(0,x) (t))W (t, φ(0,x) (t)) = g(φ(0,x) (t)) − g(φ(0,x) (t)) = Hence, F (t, φ(0,x) (t)) = f (x) for all t ∈ [0, 1], and, in consequence, f (Φt (x)) + tg(Φt (x)) = f (x), for x ∈ Ω0 This proves the theorem in the real case We next consider the complex case, i.e., K = C For each z ∈ C, z¯ stands for the complex conjugate of z; the norm of x := (x1 , , xn ) ∈ Cn is defined by kxk := p |x1 |2 + · · · + |xn |2 Let F (t, x) := f (x) + tg(x) as before and consider the following system of differential equations y (t) = W (t, y(t)), where W (t, x) := (W1 (t, x), , Wn (t, x)) with Wi , i = 1, , n, defined by  −g(x) ∂F  if Gradx F (t, x) 6= 0, |xi | ∂x (t, x) i Wi (t, x) := kGradx F (t,x)k 0 otherwise Then the rest of the theorem follows as in the real case Remark 4.7 The method used here to prove Theorem 4.2 modifies a method used in [29, 32] (see also [20, 49, 54, 58, 59]) 53 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Conclusions The main 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 in terms of Newton polyhedra with some tools of Singularity Theory and Semi-algebraic Geometry 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 (see Theorem 2.3) - 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 We present relationships between these topological equivalence types and the result is presented in Theorem 3.3 - Showing that the Lojasiewicz exponent and the multiplicity of analytic function germs are invariants of the bi-Lipschitz K-equivalence (Corollary 3.9 and Corollary 3.10) - Proving that every non-negative analytic function germ f , which satisfies Kouchnirenko’s non-degeneracy condition, is sub-analytically bi-Lipschitz C-equivalent P to the polynomial α xα , where the sum is taken over the set of all vertices of the Newton polyhedron of f (see Theorem 3.16) - 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 in Theorem 4.2 54 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn

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