Lojasiewicz inequalities, topological equivalences and newton polyhedra

j=1,...,n dj. 28 Chapter 3 The sub-analytically topological types of function germs In this chapter, we investigate 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, i.e., G = A,K, C, or V . We present relationships between these topo- logical equivalence types. In particular, for sub-analytic C1-function germs with isolated singularities the definitions of sub-analytically C0-A, C0-K, and C0-V-equivalence are equivalent. We show that the Lojasiewicz exponent and the multiplicity of analytic func- tion germs are invariants of the bi-Lipschitz K-equivalence. We also prove that every non-negative analytic function germ f , which satisfies Kouchnirenko’s non-degeneracy condition, is sub-analytically bi-Lipschitz C-equivalent (and hence, sub-analytically C0- A-equivalent) to the polynomial∑α xα, where the sum is taken over the set of all vertices of the Newton polyhedron of f. The results presented in this chapter are published in Houston Journal of Mathematics ([BP-2]). 3.1 G-equivalence A fundamental problem in Singularity Theory is the local classification of smooth maps up to diffeomorphisms. 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 . For references to the extensive literature on the subject we refer the reader to the survey of Wall [60] and the 29 papers cited therein. Since the problem of smooth classification has a lot of rigidity, it seems natural to study the classification of maps by weaker equivalence relations in which the changes of coordinates are homeomorphisms instead of diffeomorphisms. The classification problem with respect to C0-A and C0-V-equivalence relations has been well-studied. In the papers [42, 43], Nishimura investigated the classification of smooth maps up to C0-K-equivalence. This equivalence relation is the topological version of K-equivalence (or contact equivalence) introduced by Mather [35, 36]. Recently, new works have also treated such theme [1, 3, 4, 5, 13, 14, 15, 50, 56]. In this chapter, we are interested in the sub-analytically, bi-Lipschitz C0-G-equivalence of continuous sub analytic function germs from (Rn, 0) to (R, 0), where G = A,K, C, or V . We will give precise definitions of these equivalence relations at follows. Definition 3.1. Let f, g : (Rn, 0) → (R, 0) be two continuous function germs. We say that (i) f and g are called C0-A-equivalent if there exist two germs of homeomorphisms h : (Rn, 0)→ (Rn, 0) and k : (R, 0)→ (R, 0) such that the following diagram com- mutes: (Rn, 0) f−−−→ (R, 0) h y yk (Rn, 0) g−−−→ (R, 0). The function germs f and g are said to be right equivalent if they are C0-A- equivalent with h being the identity map. (ii) f and g : (Rn, 0) → (R, 0) are called C0-K-equivalent if there exist two germs of homeomorphisms h : (Rn, 0) → (Rn, 0) and H : (Rn × R, 0) → (Rn × R, 0) such that H(Rn × {0}) = Rn × {0} and the following diagram is commutative: (Rn, 0) (id,f)−−−→ (Rn × R, 0) pin−−−→ (Rn, 0) h y yH hy (Rn, 0) (id,g)−−−→ (Rn × R, 0) pin−−−→ (Rn, 0), where, id : (Rn, 0)→ (Rn, 0) is the identity map and pin : (Rn × R, 0)→ (Rn, 0) is the canonical projection. 30 The function germs f and g are said to be C0-C-equivalent if they are C0-K- equivalent with h being the identity map. (iii) f and g are called C0-V-equivalent if there exists a germ of homeomorphism h : (Rn, 0)→ (Rn, 0) such that h(f−1(0)) = g−1(0). In these definitions, if the homeomorphisms are sub-analytic (resp., bi-Lipschitz), we will say that f and g are sub-analytically (resp., bi-Lipschitz) C0-G-equivalent, where G is one of the classical Mather’s groups, i.e., G = A,K, C, or V . By definition, the following relations between these three concepts hold true: C0-A-equivalence ⇒ C0-K-equivalence ⇒ C0-V-equivalence. Remark 3.2. (i) For C∞-stable map germs, Mather in [36] proved that C0-K-equivalence implies C0-A-equivalence. (ii) For analytic function germs with isolated singularities in two or three dimensions, it is was proved by King in [26] (see also [1, 46]) that C0-V-equivalence implies C0-A- equivalence. Further, for any n ≥ 7, he constructs examples of polynomial function germs f, g : (Rn, 0) → (R, 0) with isolated singularities which are C0-V-equivalent, but not C0-A-equivalent. On the other hand, in [43], it pointed out by Nishimura that C0- V-equivalence of smooth functions with isolated singularities implies C0-K-equivalence. (iii) In the complex setting, it was shown by Saeki [51] that two holomorphic function germs from (Cn, 0) to (C, 0) with isolated singularities are C0-A-equivalent if and only if they are C0-V-equivalent. 3.2 Sub-analytically topological types The aim of this section is the comparison of C0-A, C0-K and C0-V for sub-analytic function germs (see also [3, 5, 50]). Theorem 3.3. Let f, g : (Rn, 0)→ (R, 0) be two continuous sub-analytic function germs. Then, the following are equivalent: (i) f and g are sub-analytically C0-A-equivalent. (ii) f and g are sub-analytically C0-K-equivalent. 31 (iii) There exist a germ of sub-analytic homeomorphism h : (Rn, 0)→ (Rn, 0) and some positive constants c1, c2, α, and β such that: (iii1) c1|f(x)|α ≤ |g(h(x))| ≤ c2|f(x)|β for all ‖x‖  1; and (iii2) sign[f(x)g(h(x))] is constant on the set f(x) 6= 0. (iv) There exists a germ of sub-analytic homeomorphism h : (Rn, 0) → (Rn, 0) such that: (iv1) h(f−1(0)) = g−1(0); and (iv2) sign[f(x)g(h(x))] is constant on the set f(x) 6= 0. We start with the lemma about sub-analytically PL-A-equivalence of sub-analytically PL functions, which to prove by Birbrair and Nuno-Ballesteros in [5]. Lemma 3.4. Let f1 : X1 → R and f2 : X2 → R be two sub-analytically PL functions. Assume there is a sub-analytic homeomorphism h : X1 → X2 such that h(f−11 (0)) = f−12 (0) and moreover the sign of f1(x)f2(h(x)) is constant on X1\f−11 (0). Then there are neighborhoods Ni of f −1 i (0) on Xi and Vi of 0 in R such that the restrictions fi : Ni → Vi are sub-analytically C0-A-equivalent. Proof of Theorem 3.3. (i) ⇒ (ii) and (iii) ⇒ (iv) are straightforward. (ii) ⇒ (i). We first note that since f and g are sub-analytic functions, by Theorem 1.32 they are triangulable on a small enough neighborhood of the origin. Hence we can choose triangulations (Rn, 0) f // (R, 0) (X1, 0) α1 OO f1 :: (Rn, 0) g // (R, 0) (X2, 0) α2 OO f2 :: where Xi are polyhedra, fi : Xi −→ R are sub-analytically PL-maps and αi are sub- analytic homeomorphisms. 32 Now, the hypothesis that f, g are sub-analytically C0-K-equivalent implies that there is a commutative diagram: (X1, 0) (id,f1)−−−→ (X1 × R, 0) pi1−−−→ (X1, 0) h y yH hy (X2, 0) (id,f2)−−−→ (X2 × R, 0) pi1−−−→ (X2, 0), where h,H are sub-analytic homeomorphisms, id is the identity mapping and pi1 is the projective onto the first factor. We write H(x, y) = (h(x), θx(y)), then we have that θx : (R, 0) −→ (R, 0) is a family of sub-analytic homeomorphisms depending continuously on x. In particular, we have that either: for any x, θx is always increasing, or for any x, θx is always decreasing. With this notation, the K-equivalence is written as θx(f1(x)) = f2(h(x)) ∀x ∈ X1. Then we have that h(f−11 (0)) = f −1 2 (0) and that the sign of f1(x)f2(h(x)) is constant on X1 \ f−11 (0). The result follows now from Lemma 3.4. (ii) ⇒ (iii). Suppose that the germs of the functions f and g are sub analytically C0- K-equivalent. Then, there exist germs of sub-analytic homeomorphisms h : (Rn, 0) → (Rn, 0) and H : (Rn ×R, 0)→ (Rn ×R, 0) satisfying the conditions of Definition 3.1(ii). Let V+ := {(x, y) ∈ Rn × R : y > 0} and V− := {(x, y) ∈ Rn × R : y < 0}. Since H(Rn × {0}) = Rn × {0}, we have one of the following cases: • H(V+) = V+ and H(V−) = V−; or • H(V+) = V− and H(V−) = V+. Hence, the functions f and g ◦ h (or −g ◦ h) have the same sign on each connected component of the set f(x) 6= 0. Moreover, because H is a germ of sub-analytic homeo- morphism, it follows from Lojasiewicz inequality (Theorem 1.16) that there exist some positive real numbers c1, c2, α, and β such that c1‖(x, y)− (x′, y′)‖α ≤ ‖H(x, y)−H(x′, y′)‖ ≤ c2‖(x, y)− (x′, y′)‖β, for all (x, y) and (x′, y′) in a some neighborhood of (0, 0) ∈ Rn × R. Therefore, we have 33 for all x near 0 ∈ Rn, |g(h(x))| = ‖(h(x), 0)− (h(x), g(h(x))‖ = ‖H(x, 0)−H(x, f(x))‖ ≥ c1‖(x, 0)− (x, f(x))‖α = c1|f(x)|α. Using the same argument we obtain |g(h(x))| ≤ c2|f(x)|β. (iv) ⇒ (ii). Let h : (Rn, 0)→ (Rn, 0) be a germ of sub-analytic homeomorphism satisfy- ing the conditions (iv1) and (iv2). Define the sub-analytic map germ H : (Rn×R, 0)→ (Rn × R, 0) as follows: H(x, y) :=  (h(x), 0) if y = 0, (h(x), g(h(x)) f(x) y) if 0 < y ≤ f(x) or f(x) ≤ y < 0, (h(x), y − f(x) + g(h(x))) if y ≥ f(x) ≥ 0 or y ≤ f(x) ≤ 0, (h(x), y) otherwise. Since f is continuous, a function germ F : (Rn × R, (0, 0))→ (R, 0), (x, y) 7→ f(x)− y is continuous and the set F−1(0) = {(x, y) | f(x) = y}. Thus, the map H is also contin- uous. Moreover, H is a germ of sub-analytic homeomorphism with the inverse map is given by: H−1(u, v) =  (h−1(u), 0) if v = 0, (h−1(u), f(h −1(u)) g(u) v) if 0 < v ≤ g(u) or g(u) ≤ v < 0, (h−1(u), v − g(u) + f(h−1(u))) if v ≥ g(u) ≥ 0 or v ≤ g(u) ≤ 0 (h−1(u), v) otherwise. By assumption that h(f−1(0)) = g−1(0), we can check that the germs of homeomor- phisms h and H satisfy the conditions of Definition 3.1(ii). We give now some interesting consequences of Theorem 3.3. Corollary 3.5. Let f, g : (Rn, 0)→ (R, 0), n ≥ 2, be two sub-analytic C1-function germs with isolated singularities. Then, the following are equivalent: 34 (i) f and g are sub-analytically C0-A-equivalent. (ii) f and g are sub-analytically C0-K-equivalent. (iii) f and g are sub-analytically C0-V-equivalent. Proof. (i) ⇔ (ii) follows directly from Theorem 3.3. (ii) ⇒ (iii) are straightforward. (iii) ⇒ (ii). Since f and g are sub-analytically C0-V-equivalent, there exists a germ of sub-analytic homeomorphism h : (Rn, 0) → (Rn, 0) such that h(f−1(0)) = g−1(0). Since the origin is an isolated critical point of f (and g), we can see that f (resp., g) changes its sign on each connected component of the set f(x) 6= 0 (resp., g(x) 6= 0). Hence the sign of f(x)g(h(x)) is constant on the set f(x) 6= 0. This, together with Theorem 3.3, proves (ii). Remark 3.6. Let n = 1 and f : (R, 0) → (R, 0) be a non-constant continuous sub- analytic function germ. By Monotonicity Lemma (Theorem 1.12), there exists a constant  > 0 such that f is monotone on the intervals (−, 0] and [0, ). Moreover, thanks to Lojasiewicz inequality (Theorem 1.17), there exist some positive constants c1, c2, α, and β such that c1‖x‖α ≤ |f(x)| ≤ c2‖x‖β for all ‖x‖ ≤ . It follows from Theorem 3.3 that f is sub-analytically C0-A-equivalent to the polynomial x2 (resp., x) if 0 is an extreme point (resp., 0 is not an extreme point) of f. Corollary 3.7. Let f : (Rn, 0) → (R, 0), n ≥ 2, be a continuous sub-analytic function germ. Then f−1(0) = {0} if and only if f is sub-analytically C0-A-equivalent to the polynomial ‖x‖2 := ∑ni=1 x2i . Proof. The sufficient condition is clear. Conversely, assume f−1(0) = {0}. Then f does not change its sign on a neighborhood of 0 in Rn. Without loss of generality, we can assume that f ≥ 0. By Lojasiewicz inequality (Theorem 1.17), there exist some positive constants c1, c2, α, and β such that c1‖x‖α ≤ f(x) ≤ c2‖x‖β for all ‖x‖  1. This, together with Theorem 3.3, implies the required statement. 35 Another consequence of Theorem 3.3 is as follows (see also [3, Theorem 2.4], [50, Propo- sition 3.10]). Corollary 3.8. Let f, g : (Rn, 0) → (R, 0) be two (sub-analytic) Lipschitz function germs. Then, the following statements are equivalent: (i) f and g are (sub-analytically) bi-Lipschitz K-equivalent. (ii) There exist a germ of (sub-analytic) bi-Lipschitz homeomorphism h : (Rn, 0) → (Rn, 0) and some real numbers c1, c2 > 0, and σ ∈ {±1} such that: c1f(x) ≤ σg(h(x)) ≤ c2f(x) for all ‖x‖  1. Proof. The proof is almost the same as in [3, Theorem 2.4]. (i) ⇒ (ii). Since f and g are (sub-analytically) bi-Lipschitz K-equivalent, there ex- ist germs of (sub-analytically) bi-Lipschitz homeomorphisms h : (Rn, 0) → (Rn, 0) and H : (Rn × R, 0) → (Rn × R, 0) satisfying the conditions of Definition 3.1(ii). Let V+ := {(x, y) ∈ Rn × R : y > 0} and V− := {(x, y) ∈ Rn × R : y < 0}. Since H(Rn × {0}) = Rn × {0}, we have one of the following cases: • H(V+) = V+ and H(V−) = V−; or • H(V+) = V− and H(V−) = V+. Hence, the functions f and g ◦ h (or −g ◦ h) have the same sign on each connected component of the set f(x) 6= 0. Moreover, as H is bi-Lipschitz, we have for all x near 0 ∈ Rn, |g(h(x))| = ‖(h(x), 0)− (h(x), g(h(x))‖ = ‖H(x, 0)−H(x, f(x))‖ ≥ c1‖(x, 0)− (x, f(x))‖ = c1|f(x)|. Using the same argument we obtain |g(h(x))| ≤ c2|f(x)|. The rest of the proof follows by f and g ◦ h (or −g ◦ h) having the same sign on each connected component of the set f(x) 6= 0. (ii) ⇒ (i). Since the germs f and g are Lipschitz, it is easy to check that the germ of (sub-analytic) homeomorphism H constructed in the proof of Theorem 3.3 is actually bi-Lipschitz. 36 3.3 Bi-Lipschitz K-equivalence invariances of the Lojasiewicz exponent and the multiplicity Let f : (Rn, 0)→ (R, 0) be a continuous sub-analytic function germ. The Classical Lojasiewicz inequality asserts that there exist constants δ > 0, c > 0, and l > 0 such that |f(x)| ≥ cd(x, f−1(0))l for all ‖x‖ ≤ δ, where d(x, f−1(0)) := inf{‖x − y‖ | y ∈ f−1(0)}. 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. Suppose that f is not identically 0. 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 following gives a criterion for two function germs f, g : (Rn, 0)→ (R, 0) to have the same Lojasiewicz exponents at the origin. Corollary 3.9. Let f, g : (Rn, 0) → (R, 0) be two non-constant sub-analytic Lipschitz function germs. If f and g are bi-Lipschitz K-equivalent, then L0(f) = L0(g). Proof. Using the same argument in the proof of Corollary 3.8, there exist a germ of sub-analytic bi-Lipschitz homeomorphism h : (Rn, 0) → (Rn, 0) and some real numbers c1, c2 > 0, such that: c1|f(x)| ≤ |g(h(x))| ≤ c2|f(x)| for all ‖x‖  1. In particular, we have h(f−1(0)) = g−1(0). It follows from the results of Bochnak and Risler in [8] that the Lojasiewicz exponent L0(g) is attained, i.e., there exists a constant c3 > 0 such that |g(x)| ≥ c3d(x, g−1(0))L0(g) for ‖x‖  1. Therefore, for any x ∈ Rn such that x 6∈ f−1(0), ‖x‖  1 we have d(x, f−1(0)) ≤ ‖x− y‖ for all y ∈ f−1(0) ≤ L‖h(x)− h(y)‖ for all y ∈ f−1(0). 37 Then d(x, f−1(0)) ≤ L‖h(x)− h(f−1(0))‖ ≤ Ld (h(x), g−1(0)) ≤ L ( 1 c3 ) 1 L0(g) |g(h(x))| 1L0(g) ≤ L ( c2 c3 ) 1 L0(g) |f(x)| 1L0(g) . Thus cd(x, f−1(0))L0(g) ≤ |f(x)|, for all x near the origin, with c = ( 1 L )L0(g) c3 c2 > 0. It follows the definition of L0(f) that L0(g) ≥ L0(f). Using the same argument we obtain L0(f) ≥ L0(g). So the corollary follows. Let f : (Rn, 0)→ (R, 0) be an analytic function germ, f(x) = fm(x) + fm+1(x) + · · · , with fi a homogeneous polynomial of degree i, and fm 6≡ 0. We denote by m0(f) := m, the multiplicity of f. The following result states that the multiplicity is an invariant of the bi-Lipschitz K-equivalence (see also [12, 48]). Corollary 3.10. Let f, g : (Rn, 0) → (R, 0) be two analytic function germs. If f and g are bi-Lipschitz K-equivalent, then m0(f) = m0(g). Proof. This is an application of Corollary 3.8 and [19, Proposition 2.2]. Remark 3.11. The above corollary was proved by Fernandes and Ruas [19, Corollary 2.3] under the stronger assumption that f and g are bi-Lipschitz right equivalent. 3.4 Sub-analytically bi-Lipschitz C-equivalence and non-degeneracy conditions In this section, under some Newton non-degeneracy conditions, we show that an- alytic function germs to be sub-analytically bi-Lipschitz C-equivalent to their Newton principal parts. 38 Let Γ+(f) and Γ(f), which are defined as in Subsection 1.3.1, be the Newton poly- hedron and the Newton boundary at origin of f, respectively. For each face ∆ ∈ Γ(f), we denote by f∆(x) the polynomial ∑ α∈∆ cαx α. In particular, fΓ(x) := ∑ α∈Γ(f) cαx α. We also let V (f) be the set of all vertices of Γ+(f) and put PΓ(x) := ∑ α∈V (f) |xα|. Remark 3.12. (i) It is easy to check that the function PΓ is sub-analytic and Lipschitz in a neighborhood of the origin 0 in Rn. (ii) If f ≥ 0 then all coordinates of vertices of Γ+(f) are even, and hence PΓ is a polynomial function. Theorem 3.13. Let f : (Rn, 0) → (R, 0) be an analytic function germ. If f satisfies (M-G) condition, then the following statements hold true: (i) f and PΓ are sub-analytically bi-Lipschitz C-equivalent if and only if either f ≥ 0 or f ≤ 0 on Rn. (ii) f and fΓ are sub-analytically bi-Lipschitz C-equivalent. Let us present some results before proving the above theorem. Proposition 3.14. Let f : (Rn, 0)→ (R, 0) be an analytic function germ. The following statements are equivalent: (i) f satisfies (M-G) condition. (ii) There exist some positive real numbers c1 and c2 such that: c1PΓ(x) ≤ |f(x)| ≤ c2PΓ(x) for ‖x‖  1. Proof. (i) ⇒ (ii). Suppose that f satisfies (M-G) condition. We will prove the first inequality; the same argument applies to the second inequality. By contradiction and using the Curve Selection Lemma (see Theorem 1.13), there exists an analytic curve φ : (0, )→ Rn, s 7→ (φ1(s), . . . , φn(s)), such that: (a) ‖φ(s)‖ → 0 as s→ 0+; and (b) PΓ(φ(s)) |f(φ(s))|. 39 Let J := {j |φj 6≡ 0} ⊂ {1, . . . , n}. For j ∈ J we can expand the coordinate function φj in terms of the parameter, say: φj(s) = x 0 js aj + higher order terms in s, where x0j 6= 0 and aj ∈ Q. From Condition (a), we get aj > 0. Let RJ := {α = (α1, . . . , αn) ∈ Rn |αj = 0 for j 6∈ J}. If Γ(f) ∩ RJ = ∅ then for any α ∈ Γ(f), there exists an index j /∈ J such that αj > 0 and so (φj(s))αj ≡ 0. Then, (φ(s))α = n∏ j=1 (φj(s)) αj = ∏ j∈J (φj(s)) αj ∏ j 6∈J (φj(s)) αj ≡ 0. Hence PΓ(φ(s)) = ∑ α∈V (f) |φ(s)α| ≡ 0, which contradicts to Condition (b). Therefore, Γ(f) ∩ RJ 6= ∅. Let d be the minimal value of the linear function ∑j∈J ajαj on Γ(f)∩RJ , i.e., d := minα∈Γ(f)∩RJ ∑ j∈J ajαj and let ∆ be the (unique) maximal face of Γ(f) where the linear function ∑ j∈J ajαj takes its minimal value; i.e., ∆ = {α ∈ Γ(f) ∩ RJ | ∑ j∈J ajαj = d}. By definition, then d > 0 and ∆ ∈ Γ(f). We can write f(φ(s)) = f∆(x 0)sd + higher order terms in s, where x0 := (x01, . . . , x 0 n) and x 0 j := 1 for j 6∈ J. Since f satisfies (M-G) condition, f∆(x 0) 6= 0, and hence f(φ(s)) ' sd as s→ 0+. On the other hand, a simple computation shows that PΓ(φ(s)) = ∑ α∈V (f) |φ(s)α| = ∑ α∈∆ |(x0)α|sd + higher order terms in s. Therefore PΓ(φ(s)) ' f(φ(s)) ' sd as s→ 0+. 40 This combined with (b) gives a contradiction. (ii) ⇒ (i). By contradiction, suppose that there exists a face ∆ ∈ Γ(f) and a point x0 ∈ (R \ {0})n such that f∆(x0) = 0. Let J be the smallest subset of {1, . . . , n} such that ∆ ⊂ Γ(f)∩RJ . Then, there exists a non-zero vector a ∈ Rn+, with aj > 0 for j ∈ J, such that ∆ = {α ∈ Γ(f) ∩ RJ | 〈a, α〉 = min β∈Γ(f)∩RJ 〈a, β〉}. Let φ(s) : (0, ε)→ Rn, s 7→ (φ1(s), . . . , φn(s)), be the monomial curve given by φj(s) = { x0js aj if j ∈ J, 0 otherwise. Let d := minα∈Γ(f)∩RJ 〈a, α〉. We can write f(φ(s)) = f∆(x 0)sd + higher order terms in s. Since f∆(x 0) = 0, it follows that f(φ(s)) = o(sd) as s→ 0+. (3.1) On the other hand, we have PΓ(φ(s)) = ∑ α∈V (f) |φ(s)α| = ∑ α∈∆ |(x0)α|sd + higher order terms in s. Consequently, PΓ(φ(s)) ' sd as s→ 0+. (3.2) By assumption, we have PΓ(φ(s)) ' |f(φ(s))| as s→ 0+. This, together with (3.1) and (3.2), gives a contradiction. Proposition 3.15. Let f : (Rn, 0) → (R, 0) be an analytic function germ satisfying (M-G) condition. Then, the following statements hold true (i) f satisfies (K) condition. (ii) There exist some positive constants c1 and c2 such that: c1fΓ(x) ≤ f(x) ≤ c2fΓ(x) for ‖x‖  1. 41 Proof. (i). Let ∆ be any face in Γ(f). By definition, the polynomial f∆ is quasi homo- geneous, i.e., there exist a non-zero vector a := (a1, . . . , an) ∈ Zn≥0 and a positive integer d such that f∆(t a1x1, . . . , t anxn) = t df∆(x1, . . . , xn) for all numbers t ∈ R and all points x := (x1, . . . , xn) ∈ Rn. Taking the t-derivative of both sides and then letting t = 1, we obtain n∑ j=1 ajxj ∂f∆(x) ∂xj = d f∆(x). Hence, if ∂f∆(x) ∂xj = · · · = ∂f∆(x) ∂xn = 0 for some x ∈ (R \ {0})n, then f∆(x) = 0, which contradicts to the assumption. (ii). We will prove the first inequality; the second inequality can be proved similarly. By contradiction and using the Curve Selection Lemma, there exists an analytic curve φ : (0, )→ Rn, s 7→ (φ1(s), . . . , φn(s)), such that: (a) ‖φ(s)‖ → 0 as s→ 0+; and (b) fΓ(φ(s)) f(φ(s)). Let J := {j |φj 6≡ 0} ⊂ {1, . . . , n}. For j ∈ J we can expand the coordinate φj in terms of the parameter, say: φj(s) = x 0 js aj + higher order terms in s, where x0j 6= 0 and aj ∈ Q. From Condition (a), we get aj > 0. Recall that RJ := {α = (α1, . . . , αn) ∈ Rn |αj = 0 for j 6∈ J}. If Γ(f) ∩ RJ = ∅ then for any α ∈ Γ(f), there exists an index j /∈ J such that αj > 0 and so (φj(s))αj ≡ 0. Then, (φ(s))α = n∏ j=1 (φj(s)) αj = ∏ j∈J (φj(s)) αj ∏ j 6∈J (φj(s)) αj ≡ 0. Hence fΓ(φ(s)) = ∑ α∈Γ(f) cα (φ(s)) α ≡ 0, which contradicts to Condition (b). 42 Therefore, Γ(f) ∩ RJ 6= ∅. Let d be the minimal value of the linear function ∑j∈J ajαj on Γ(f)∩RJ , i.e., d := minα∈Γ(f)∩RJ ∑ j∈J ajαj and let ∆ be the (unique) maximal face of Γ(f) where the linear function ∑ j∈J ajαj takes its minimal value; i.e., ∆ = {α ∈ Γ(f) ∩ RJ | ∑ j∈J ajαj = d}. By definition, then d > 0 and ∆ ∈ Γ(f). We can write f(φ(s)) = f∆(x 0)sd + higher order terms in s, where x0 := (x01, . . . , x 0 n) and x 0 j := 1 for j 6∈ J. Since f satisfies (M-G) condition, we get f∆(x 0) 6= 0, and so f(φ(s)) ' sd as s→ 0+. (3.3) On the other hand, it is easy to see that fΓ(φ(s)) = f∆(x 0)sd + higher order terms in s. This, together with (3.3) and (b), gives a contradiction. Proof of Theorem 3.13. (i). Assume that f and PΓ are sub-analytically bi-Lipschitz C- equivalent. Then, there exists a germ of sub-analytic homeomorphism H : (Rn×R, 0)→ (Rn × R, 0) satisfying the conditions of Definition 3.1(ii). By the same argument in the proof of Corollary 3.8, there exist some positive constants c1, c2 such that c1PΓ(x) ≤ σ|f(x)| ≤ c2PΓ(x), for all ‖x‖  1. Hence we have either f ≥ 0 or f ≤ 0. The sufficient follows immediately from Proposition 3.14 and Corollary 3.8. (ii). Thanks to Proposition 3.15, there exist some positive constants c1 and c2 such that: c1fΓ(x) ≤ f(x) ≤ c2fΓ(x) for ‖x‖  1. This, together with Corollary 3.8, yields (ii). Theorem 3.16. Let f : (Rn, 0)→ (R, 0) be an analytic function germ such that f ≥ 0. Then, the following statements are equivalent: (i) f satisfies (M-G) condition. 43 (ii) f satisfies (K) condition. (iii) For any face ∆ ∈ Γ(f) we have f∆ > 0 on (R \ {0})n. (iv) There exist some positive constants c1 and c2 such that: c1PΓ(x) ≤ f(x) ≤ c2PΓ(x) for all ‖x‖  1. (v) f and PΓ are sub-analytically bi-Lipschitz C-equivalent. Proof of Theorem ??. (i) ⇒ (ii). This is straightforward from Proposition 3.15. (ii) ⇒ (i). By Lemma 2.4, we have f∆(x) ≥ 0 on Rn. Suppose that there exists a point x0 ∈ (R \ {0})n such that f∆(x0) = 0. Then x0 is a global minimal point of f∆. Consequently, x0 is a critical point of f∆, which is a contradiction because f satisfies (K) condition. (i) ⇒ (iii). This is a direct consequence of Lemma 2.4. (iii) ⇒ (i). This is obvious. (i) ⇔ (iv). This follows immediately from Proposition 3.14. (iv) ⇔ (v). This is evident from Corollary 3.8. As consequences, we have the following results. Corollary 3.17. Let f : (Rn, 0) → (R, 0) be an analytic function germ. If f satisfies (M-G) condition, then f and fΓ are sub-analytically C 0-A-equivalent. Proof. This follows immediately from Theorems 3.3 and 3.13. Corollary 3.18. Let f : (Rn, 0) → (R, 0) be a non-negative analytic function germ. If f satisfies (K) condition, then f and PΓ are sub-analytically C 0-A-equivalent. Proof. This follows immediately from Theorems 3.3 and 3.16. Remark 3.19. Fukui and Yoshinaga [20] (see also [59]) have proved that an analytic function germ f satisfying (K) condition is C0-A-equivalent to its Newton principal part fΓ; but, as their proof involves integration of vector fields, it does not provide a sub-analytically C0-A-equivalence. 44 Chapter 4 Analytically principal part of polynomials at infinity Let f : Kn → K be a polynomial function, where K := R or C. In this chapter we give, in terms of the Newton boundary at infinity of f, a sufficient condition for a deformation of f to be analytically (smooth in the case K := C) trivial at infinity. This chapter is written based on our paper [BP-3]. 4.1 Introduction The problem of C0-sufficiency of jets is one of the most interesting problems in Singularity Theory. Recall that the k-jet of a Cr-function in the neighborhood of 0 ∈ Rn is identified with its k-th Taylor polynomial of at 0, then the function is called a realization of the jet. The k-jet is said to be Cp-sufficient in the Cr class (p ≤ r), if for any two of its Cr-realizations f and g, there exists a Cp-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 following: Let f : Rn → R be a Ck in a neighborhood of 0 ∈ Rn satisfying 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. 45 (ii) The k-jet of f is sufficient in the Ck 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-Nogue`s and Ha` [10]: Let f be a polynomial of C[z1, z2]. Then, two following conditions are equiv- alent: (i) There are positive constants C and R such that ‖∇f(x)‖ ≥ C‖x‖k−1 for ‖x‖ ≥ R. (ii) There exist a positive constant  such that for every polynomial P ∈ C[z1, z2] of degree less or equal k, whose modules of coefficients of mono- mials 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 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-Nogue`s and Ha` is recently generalised by Skalski [54], Rodak and Spodzieja [49]. On the other hand, except for certain degenerate cases the topological type of an analytic function is expected to depend only on its Newton polyhedron. This has been confirmed together with a precise definition of non-degeneracy at the origin of Kn (see, for example, [16, 17, 20, 28, 44, 59]). The purpose of this chapter is to show that polynomial functions, which are con- venient and non-degenerate at infinity, are determined up to analytical type by their Newton polyhedra at infinity. 4.2 Main Theorem Every open set of the form Kn \ K, where K ⊂ Kn is a compact set, is called a neighborhood of infinity. 46 Definition 4.1. Let f, g : Kn → K be polynomial functions. We say that the family f + tg is analytically (resp., smooth) trivial at infinity along the interval [0, 1] if there exist a neighborhood of infinity Ω0 ⊂ Kn and a continuous mapping Φ: [0, 1] × Ω0 → Kn, (t, x) 7→ Φ(t, x), such that the following conditions are satisfied (a) Φ0(x) = x for x ∈ Ω0; (b) for any t ∈ [0, 1], the mapping Φt : Ω0 → Φt(Ω0) is a real analytic diffeomorphism (resp., C∞-diffeomorphism) and lim x→∞ Φt(x) =∞; (c) f(Φt(x)) + tg(Φt(x)) = f(x) for x ∈ Ω0 and t ∈ [0, 1]; where the mapping Φt : Ω0 → Kn is defined by Φt(x) := Φ(t, x) for x ∈ Ω0 and t ∈ [0, 1]. With the above definitions, the main result is as follows. Theorem 4.2. Let f : Kn → K be a polynomial function. Suppose that f is convenient and non-degenerate at infinity. Then, for any polynomial function g : Kn → K satisfying the condition Γ−(g) ⊂ Int(Γ−(f)), the family f+ tg is analytically (resp., smooth) trivial at infinity along the interval [0, 1] in the case K = R (resp., K = C). Let us recall that Γ−(f) and Γ∞(f) be a Newton polyhedron and a Newton boundary at infinity of f , respectively. We will prove Theorem 4.2 in the case K = R; the proof is quite similar in the complex case K = C. So, let f, g : Rn → R be polynomial functions satisfying the conditions of Theorem 4.2. Let us consider the polynomial function F : R× Rn → R, (t, x) 7→ f(x) + tg(x). For any fixed t ∈ R, let ft(x) := F (t, x). By the assumption Γ−(g) ⊂ Int(Γ−(f)), we have Γ−(f) ≡ Γ−(ft) and Γ∞(f) ≡ Γ∞(ft). Furthermore, ft is convenient and non-degenerate at infinity. For simplicity, we let GradxF (t, x) := ( x1 ∂F ∂x1 (t, x), . . . , xn ∂F ∂xn (t, x) ) . Lemma 4.3. There exist positive constants C and R such that ‖GradxF (t, x)‖ ≥ C|g(x)| ∀ t ∈ (−2, 2),∀ ‖x‖ > R. 47 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) ‖ϕ(s)‖ → ∞ as s→ 0+; and (c) ‖GradxF (t(s), ϕ(s))‖2 = ∑n i=1 ( ϕi(s) ∂F ∂xi (t(s), ϕ(s)) )2  |g(ϕ(s))|2. 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) = x 0 i s ai + higher order terms in s, where x0i 6= 0 and ai ∈ Q. From Condition (b), we obtain mini∈I ai < 0. Let RI := {α = (α1, . . . , αn) ∈ Rn |αi = 0 for i 6∈ I}. Since f is convenient, Γ−(f)∩RI 6= ∅. We put a := (a1, . . . , an) ∈ Rn, where ai := 0 for i 6∈ I. Let d be the minimal value of the linear function 〈a, α〉 = ∑i∈I aiαi on Γ−(f)∩RI , and let ∆ be the (unique) maximal face of Γ−(f) where the linear function ∑ i∈I aiαi takes its minimal value d, i.e., ∆ := {α ∈ Γ−(f) | 〈a, α〉 = d}. We have d < 0 and ∆ is a closed face of Γ∞(f) because f is convenient and mini=1,...,n ai < 0. Since Γ−(g) ⊂ Int(Γ−(f)) and t(s) is bounded, we can see that ϕi(s) ∂F ∂xi (t(s), ϕ(s)) = x0i ∂f∆ ∂xi (x0)sd + higher order terms in s, for i ∈ I, where x0 := (x01, . . . , x 0 n) and x 0 i := 1 for i 6∈ I. Consequently, we obtain ‖GradxF (t(s), ϕ(s))‖2 = n∑ i=1 ( ϕi(s) ∂F ∂xi (t(s), ϕ(s)) )2 = ∑ i∈I ( ϕi(s) ∂F ∂xi (t(s), ϕ(s)) )2 = s2d ∑ i∈I ( x0i ∂f∆ ∂xi (x0) )2 + higher order terms in s, = s2d n∑ i=1 ( x0i ∂f∆ ∂xi (x0) )2 + higher order terms in s. 48 (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 n∑ i=1 ( x0i ∂f∆ ∂xi (x0) )2 6= 0. Therefore ‖GradxF (t(s), ϕ(s))‖ ' 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 ‖GradxF (t, x)‖ ≥ C, ∀ t ∈ (−2, 2),∀ ‖x‖ > 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) ‖ϕ(s)‖ → ∞ as s→ 0+; and (c) ‖GradxF (t(s), ϕ(s))‖2 = ∑n i=1 ( ϕi(s) ∂F ∂xi (t(s), ϕ(s)) )2 → 0 as s→ 0+. 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) = x 0 i s ai + higher order terms in s, where x0i 6= 0 and ai ∈ Q. From Condition (b), we obtain mini∈I ai < 0. Let RI := {α = (α1, . . . , αn) ∈ Rn |αi = 0 for i 6∈ I}. Since the polynomial f is convenient, we have Γ−(f) ∩ RI 6= ∅. We put a := (a1, . . . , an) ∈ Rn, where ai = 0 for i 6∈ I. Let d be the minimal value of the linear function ∑i∈I aiαi on Γ−(f) ∩ RI , and let ∆ be the (unique) maximal face of Γ−(f) where the linear function ∑ i∈I aiαi takes 49 its minimal value d. Then it is easy to see that d < 0 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 ∂xi (t(s), ϕ(s)) = x0i ∂f∆ ∂xi (x0)sd + higher order terms in s, for i ∈ I, where x0 := (x01, . . . , x 0 n) and x 0 i := 1 for i 6∈ I. Then it follows from Condition (c) that x0i ∂f∆ ∂xi (x0i ) = 0, for i ∈ I. Note that f∆ does not depend on xi for i 6∈ I. Therefore, x0i ∂f∆ ∂xi (x0i ) = 0, 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 Wi(t, x) :=  −g(x) ‖GradxF (t,x)‖2 x 2 i ∂F ∂xi (t, x) if GradxF (t, x) 6= 0, 0 otherwise. Then there exist positive constants C and R such that the mapping W : (−2, 2)× {x ∈ Rn | ‖x‖ > R} → Rn, (t, x) 7→ (W1(t, x), . . . ,Wn(t, x)), is analytic and satisfies the following conditions ‖W (t, x)‖ ≤ C‖x‖, dxF (t, x)W (t, x) = −g(x), where dxF 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 ‖x‖ > R, ‖W (t, x)‖ ≤ |g(x)|‖GradxF (t, x)‖2 n∑ i=1 |xi| ∣∣∣∣xi ∂F∂xi (t, x) ∣∣∣∣ ≤ |g(x)|‖GradxF (t, x)‖2 √√√√ n∑ i=1 x2i √√√√ n∑ i=1 ( xi ∂F ∂xi (t, x) )2 = |g(x)| ‖GradxF (t, x)‖‖x‖. 50 By Lemma 4.3, we conclude that ‖W (t, x)‖ ≤ ‖x‖ C . Finally, we have for t ∈ (−2, 2) and ‖x‖ > R, dxF (t, x)W (t, x) = n∑ i=1 ∂F ∂xi (t, x)Wi(t, x) = n∑ i=1 −g(x)∑n k=1 ( xk ∂F ∂xk (t, x) )2 (xi ∂F∂xi (t, x) )2 = −g(x). This completes the proof of the lemma. Lemma 4.6. (See [54, Lemmas 1 and 2]) Let D := {x ∈ Rn | ‖x‖ > R}, R > 0, and let W : (−2, 2)×D → Rn be a continuous mapping such that for some C > 0 we have ‖W (t, x)‖ ≤ C‖x‖ 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 0 ∈ (α, β), h(0) = x, and ‖x‖ > ReC , then β > 1 and e−C‖x‖ ≤ ‖h(1)‖ ≤ eC‖x‖. If 1 ∈ (α, β), h(1) = x, and ‖x‖ > ReC , then α < 0 and e−C‖x‖ ≤ ‖h(0)‖ ≤ eC‖x‖. 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 | ‖x‖ > R} → Rn defined in Lemma 4.5 is analytic. Let us consider the following system of differential equations y′(t) = W (t, y(t)). (4.2) 51 Let D := {x ∈ Rn | ‖x‖ > 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), ‖x‖ > R, t ∈ I(s, x)} → Rn (s, x, t) 7→ φ(s,x)(t) is analytic. Set Ω := {x ∈ Rn | ‖x‖ > 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‖x‖ ≤ ‖Φ(t, x)‖ ≤ eC‖x‖, for t ∈ [0, 1], x ∈ Ω, (4.3) e−C‖y‖ ≤ ‖Ψ(t, y)‖ ≤ eC‖y‖, for t ∈ [0, 1], y ∈ Ω. Let Ω0 := {x ∈ Rn | ‖x‖ > 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 | ‖y‖ > 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−1t . Moreover, by Inequality (4.3), we have ‖Φt(x)‖ → ∞ if, and only if, ‖x‖ → ∞. 52 Finally, thanks to Lemma 4.5, we obtain dF dt (t, φ(0,x)(t)) = g(φ(0,x)(t)) + dxF (t, φ(0,x)(t))φ ′ (0,x)(t) = g(φ(0,x)(t)) + dxF (t, φ(0,x)(t))W (t, φ(0,x)(t)) = g(φ(0,x)(t))− g(φ(0,x)(t)) = 0. 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 ‖x‖ :=√|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 Wi(t, x) :=  −g(x) ‖GradxF (t,x)‖2 |xi|2 ∂F∂xi (t, x) if GradxF (t, x) 6= 0, 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 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 Kouch- nirenko’s non-degeneracy condition, is sub-analytically bi-Lipschitz C-equivalent to the polynomial ∑ α x α, 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 List of Author’s Related Papers [BP-1] N. T. N. Bu`i and T. S. Pha.m, Computation of the Lojasiewicz exponent of non- negative and nondegenerate analytic functions, Internat. J. Math, 25 (10) (2014), 1450092 (13 pp.). [BP-2] N. T. N. Bu`i and T. S. Pha.m, On the subanalytically topological types of function germs, Houston J. Math., 42 (4) (2016), 1111–1126. [BP-3] N. T. N. Bu`i and T. S. Pha.m, Analytically principal part of polynomials at infinity, Ann. Polon. Math., 117 (3) (2016), 259–268. 55 Bibliography [1] S. Alvarez, L. Birbrair, J. C. F. Costa, and A. Fernandes, Topological K-equivalence of analytic function germs, Cent. Eur. J. Math., 8 (2) (2010), 338–345. [2] E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Publ. Math. I.H.E.S., Bures-sur-Yvette, France, 67 (1988), 5–42. [3] L. Birbrair, J. C. F. Costa, A. Fernandes, and M. A. S. Ruas, K-bi-Lipschitz equiv- alence of real function-germs, Proc. Amer. Math. Soc., 135 (4) (2007), 1089–1095. [4] L. Birbrair, J. C. F. Costa, and A. Fernandes, Finiteness theorem for topological contact equivalence of map germs, Hokkaido Math. J., 38 (2009), 511–517. [5] L. Birbrair and J.J Nuno-Balesteros, Topologocal K and A equivalence of polynomial functions, Journal of Sing., 6 (2012), 15–18. [6] J. Bochnak and W. Kucharz, Sur les germes d’applications differentiables a` singu- larite´s isole´es, Trans. Amer. Math. Soc., 252 (1979), 115–131. [7] J. Bochnak and S. Lojasiewicz, A converse of the Kuiper-Kuo theorem, in: Pro- ceedings of Liverpool Singularities- Symposium, I, 1969/1970, in: Lecture Notes in Math., Springer, Berlin, 192 (1971), 254–261. [8] J. Bochnak and J. J. Risler, Sur les exposant de Lojasiewicz, Comment. Math. Helv., 50 (1975), 493–507. [9] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Springer, 36 (1998). [10] P. Cassou-Nogue`s and H. V. Ha`, Theoreme de Kuiper–Kuo–Bochnak– Lojasiewicz a l’infini, Ann. Fac. Sci. Toulouse Math., 5 (3) (1996), 387–406. 56 [11] S.H. Chang and Y.C. Lu, On C0-sufficiency of complex jets, Canad. J. Math., 25 (1973), 874–880. [12] G. Comte, P. Milman, and D. Trotman, On Zariski’s multiplicity problem, Proc. Amer. Math. Soc., 130 (7) (2002), 2045–2048. [13] J.C.F. Costa, A note on topological contact equivalence, in Real and Complex Sin- gularities, Lond. Math. Soc. Lecture Notes Series, 380 (2010), 114–124. [14] J. C. F. Costa and J.J Nuno-Balesteros, Topological K-classification of finitely de- termined map germs, Geom. Dedicata, 166 (2013), 147–162. [15] J. C. F. Costa, T. Nishimura, and M. A. S. Ruas, Bi-Lipschitz A-equivalence of K-equivalent map-germs, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 108 (1) (2014), 173–182 . [16] J. Damon and T. Gaffney, Topological triviality of deformations of functions and Newton filtrations, Invent. Math., 72 (1983), 335–358. [17] J. Damon, Topological equivalence for nonisolated singularities and global affine hypersurfaces, A.M.S. Contemp. Math., 90 (1989), 21–53. [18] S. T. D- inh, H. V. Ha`, T. S. Pha.m, and N. T. Tha˙’o, Global Lojasiewicz-type in- equality for nondegenerate polynomial maps, J. Math. Anal. Appl., 410 (2) (2014), 541–560. [19] A. C. G. Fernandes and M. A. R. Ruas, Bi-Lipschitz determinacy of quasihomoge- neous germs, Glasg. Math. J., 46 (1) (2004), 77–82. [20] T. Fukui and E. Yoshinaga, The modified analytic trivialization of family of real analytic functions, Invent. Math., 82 (3) (1985), 467–477. [21] H. V. Ha` and H. D. Nguye˜ˆn, Lojasiewicz inequality at infinity for polynomials in two real variables, Math. Z., 266 (2) (2010), 243–264. [22] R. M. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math., 38 (3) (1977), 207–217. [23] M. Hirsch and S. Smale, Differential equations, Dynamical Systems and Linear Algebra, Academic Press, (1974). 57 [24] S. G. Gindikin, Energy estimates connected with the Newton polyhedron, (Russian) Trudy Moskovskogo Matematicheskogo Obshchestva, 31, 189–236. (English) Trans. Moscow Math. Soc., 31 (1974), 193–246. [25] J. Gwoz´dziewicz, The Lojasiewicz exponent of an analytic function at an isolated zero, Comment. Math. Helv., 74 (3) (1999), 364–375. [26] H.C. King, Real analysis germs and their varieties isolated singularities, Invent. Math., 37 (1976), 193–199. [27] J. Kollar, An effective Lojasiewicz inequality for real polynomials, Period. Math. Hungar., 38 (3) (1999), 213–221. [28] A. G. Kouchnirenko, Polyhedres de Newton et nombre de Milnor, Invent. Math., 32 (1976), 1–31. [29] T. C. Kuo, On C0-sufficiency of jets of potential functions, Topology, 8 (1969), 167–171. [30] T. C. Kuo, Characterizations of v-sufficiency of jets, Topology, 11 (1972), 115–131. [31] T. C. Kuo, Computation of Lojasiewicz exponent of f(x, y), Comment. Math. Helv., 49 (1974), 201–213. [32] N.H. Kuiper, C1-equivalence of functions near isolated critical points, in: Sympo- sium on Infinite-Dimensional Topology, Louisiana State Univ., Baton Rouge, La., (1967), in: Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 69 (1972), 199–218. [33] K. Kurdyka and S. Spodzieja, Separation of real algebraic sets and the Lojasiewicz exponent, Proc. Amer. Math. Soc., 142 (9) (2014), 3089–3102. [34] S. Lojasiewicz, Ensembles semi-analytiques, IInst. Hautes E´tudes Sci. Publ. Math., Bures-sur-Yvette, (1965). [35] J. N. Mather, Stability of C∞-mappings, III. Finitely determined map-germs, Inst. Hautes E´tudes Sci. Publ. Math., 35 (1969), 127–156. [36] J. N. Mather, Stability of C∞ mappings. IV. Classification of stable germs by R- algebras, Inst. Hautes E´tudes Sci. Publ. Math., 37 (1969), 223–248. 58 [37] V. P. Mikhalov, The behaviour at infinity of a class of polynomials, (Russian) Trudy Mat. lnst. Steklov., 91 (1967), 59–80. [38] J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Stud., 61, Princeton University Press, (1968). [39] R. Narasimhan, Analysis on real and complex manifolds, North-Holland Math. Li- brary, 35, North-Holland Publishing Co., Amsterdam, (1985), reprint of the 1973 edition. [40] A. Ne´methi and A. Zaharia, Milnor fibration at infinity, Indag. Math., 3 (1992), 323–335. [41] J. Nie and M. Schweighofer, On the complexity of Putinar’s Positivstellensatz, J. of Comp., 23 (1) (2007), 135–150. [42] T. Nishimura, Topological types of finitely C0-K-determined map germs, Trans. Amer. Math. Soc., 312 (1989), 621–639. [43] T. Nishimura, Topological K-equivalence of smooth map-germs, Stratifications, Sin- gularities and Differential Equations, I (Marseille 1990, Honolulu, HI 1990) 82–93, Travaux en Cours, 54, Hermann, Paris (1997). [44] M. Oka, On the bifurcation of the multiplicity and topology of the Newton boundary, J. Math. Soc. Japan, 31 (3) (1979), 435–450. [45] M. Oka, Non-degenerate complete intersection singularity, Actualites Mathema- tiques, Hermann, Paris, (1997). [46] A. O. Prishlyak, Topological equivalence of smooth functions with isolated critical points on a closed surface, Topology Appli., 119 (2002), 257–267. [47] T. S. Pha.m, An explicit bound for the Lojasiewicz exponent of real polynomials, Kodai Math. J., 35 (2012), 311–319. [48] J. J. Risler and D. Trotman, Bi-Lipschitz invariance of the multiplicity, Bull. Lon- don. Math. Soc., 29 (1997), 200–204. [49] T. Rodak and S. Spodzieja, Equivalence of mappings at infinity, Bull. Sci. Math., 136 (6) (2012), 679–686. 59 [50] M.A.S. Ruas and G. Vallete, C0 and bi-Lipschitz K-equivalence of mappings, Math. Z., 269 (1-2) (2011), 293–308. [51] O. Saeki, Topological types of complex isolated hypersurface singularities, Kodai Math. J., 12 (1) (1989), 23–29. [52] M. Schweighofer, On the complexity of Schmu¨dgen’s Positivstellensatz, Journal of Comp., 20 (2004), 529–543 . [53] M. Shiota, Geometry of subanalytic and semianalytic sets, Progress in Mathematics, 5 (1997). [54] G. Skalski, On analytic equivalence of functions at infinity, Bull. Sci. Math., 135 (5) (2011), 517–530. [55] B. Teissier, Varie´te´s polaires, I. Invariants polaires des singularite´s d’hypersurfaces, Invent. Math., 40 (3) (1977), 267–292. [56] M. Takahashi, A sufficient condition that contact equivalence implies right equiva- lence for smooth function germs, Houston J. Math., 35 (3) 2009, 829–833. [57] L.van den Dries, Tame Topology and o-minimal Structures, LMS Lecture Notes, Cambridge University Press, (1997). [58] E. Yoshinaga, The modified analytic trivialization of real analytic families via blowing-ups, J. Math. Soc. Japan, 40 (2) (1988), 161–179. [59] E. Yoshinaga, Topologically principal part of analytic functions, Trans. Amer. Math. Soc., 314 (2) (1989), 803–814. [60] C. T. C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc., 13 (6) (1981), 481–539. 60 Table of Notations Cp p-times continuously differentiable mappings Z set of integer numbers Z+ set of non-negative integer numbers R set of real numbers R+ set of non-negative real numbers C set of complex numbers 〈x, y〉 canonical inner product |x| modulus, absolute value of x ‖x‖ Euclidean norm of a vector x Γ+(f) Newton polyhedron of f at the origin Γ(f) Newton boundary of f at the origin Γ−(f) Newton polyhedron of f at infinity Γ∞(f) Newton boundary of f at infinity L0(f) Lojasiewicz exponent of f at the origin 61 Index convenient, 10, 11, 26 Curve Selection Lemma, 5, 19, 39 – at infinity, 5, 48 equivalent C0-A- –, 30, 31 C0-C- –, 31 C0-K- –, 30, 31, 37 C0-V- –, 31 face, 10, 11, 16 Lojasiewicz – exponent, 15, 18 – inequality, 6, 8 – exponent, 37 – inequality, 33, 35 Classical – inequality, 6, 15 multiplicity, 38 Newton polyhedron – at infinity, 11, 47 – at the origin, 10, 16, 29, 39 non-degenerate Kouchnirenko –, 11, 16–18, 41 Kouchnirenko – at infinity, 12, 47 Mikhailov–Gindikin –, 11 Mikhailov-Gindikin –, 39 semi-algebraic, 1 semi-analytic, 7 sub-analytic, 8 triangulation, 9, 32 vertice, 10, 12, 18 62