ω r ω f ∈ ε(n) Diff r (n) r f ∈ ε(n) ω Diff r (n) 1. Introduction “ f 0 0 f 0 ” [5] [2] f ∈ ε(n) ω Diff (n) ω K| x 1 | x 1 K  0 {x 1 = 0} f ∈ ε(n) ω Diff r (n) ω f ∈ ε(n) ω Diff r (n) 2. Preliminaries ε(n, p) R n R p ε(n) ε(n, 1) r ∈ N, 0 ≤ r ≤ n I = {1, , r} 1 - Rec eived 30/09/2005, in revised form 20/01/2006. X i {(x 1 , x 2 , , x n ) ∈ R n : x i = 0} i ∈ I P (I) I X = {X σ : X σ =  i∈σ X i , X φ = IR n , σ ∈ P(I)} N o Diff r (n) = {Φ : (R n , 0) → (R n , 0) : Φ(X σ ) = X σ , σ ∈ P(I)}, Φ 2.1. Definition. s f g ∈ ε(n) s D γ f(0) = D γ g(0) γ ∈ N o | γ |≤ s s f s f J s (f) s ε(n) J s n J s J s (f) 2.2. Definition. s f ∈ ε(n) s Diff r (n) g ∈ ε(n) J s (f) = J s (g) Φ ∈ Dif f r (n) g = f ◦ Φ 2.3. Definition. f ∈ ε(n) ω Diff r (n) g ∈ ε(n) J ∞ (f) J ∞ (g) Φ ∈ Dif f r (n) g = f ◦ Φ 3. The main results 3.1. Theorem. f : (R n , 0) → (R, 0) R n R (a) f ω Diff r (n) (b) f ω Diff (n) f  f | X σ f X σ ω Dif f(n) ε(n−| σ |) | σ | σ (c) m ∞ n ⊂< x 1 ∂f ∂x 1 , . . . , x r ∂f ∂x r , ∂f ∂x r+1 , . . . , ∂f ∂x n > ε(n) (d) C, α, δ > 0 r  i=1 x 2 i ( ∂f ∂x i ) 2 + n  i=r+1 ( ∂f ∂x i ) 2 ≥ C x  α ,  x < δ. Proof. ⇒ f ∈ ε(n) f ω Diff r (n) g ∈ ε(n) J ∞ (f) J ∞ (g) Φ ∈ Dif f(n) f = g ◦ Φ. (3.1) f ω Diff r (n) g ∈ ε(n) J ∞ (f) J ∞ (g) Φ ∈ Diff r (n) g = f ◦ Φ Φ ∈ Dif f(n) f ω Diff (n) f  = f | X σ ω Diff (n) ε(n− | σ |) | σ |= k k ≤ r f  = f| x 1 = =x k =0 ω Diff r (n) ε(n−k) x = (x 1 , , x k ) x  = (x k+1 , , x n ) f  (x  ) = f  (x k+1 , , x n ) = f (0, , 0, x k+1 , , x n ) g ∈ ε(n−k) J ∞ (f)(x  ) J ∞ (g)(x  ) g(x  ) = g(x k+1 , , x n ) f  (x  ) = f(0, x) g(x, x  ) = f(x, x  ) + ϕ(x  ) ϕ(x  ) = g(x  ) − f  (x  ). (3.2) g ∈ ε(n) g(x k+1 , , x n ) ∈ ε(n−k) J ∞ f  (x  ) = J ∞ g(x  ) ϕ(x  ) ∈ m ∞ n−k J ∞ (g) = J ∞ (f) Φ(x, x  ) ∈ Dif f r (n) f(x, x  ) = g ◦ Φ(x, x  ) f(0, x  ) = g ◦ Φ(0, x  ) = f (Φ(0, x  )) + g(Φ(0, x  )) − f  (Φ(0, x  )) = g(Φ(0, x  )). f(Φ(0, x  )) = f  (Φ(0, x  )) f(0, x  ) = f  (x  ) f  (x  ) = g ◦ Φ(0, x  ) f  (x  ) ω Diff (n) ε(n−k) ⇒ f ω Dif f(n) ε(n) m ∞ n ⊂< ∂f ∂x 1 , , ∂f ∂x n > ε(n), (3.3) f  = f | x 1 = =x k =0 = f (0, x  ) ω Diff (n) ε(n−k) 0 ≤ k ≤ r m ∞ n−k ⊂< ∂(0, x  ) ∂x k+1 , , ∂(0, x  ) ∂x n > ε(n − k). (3.4) ϕ(x, x  ) ∈ m ∞ n ϕ(x, x  ) ∈< x 1 ∂f ∂x 1 , , x r ∂f ∂x r , ∂f ∂x r+1 , , ∂f ∂x n > ε(n). (3.5) ϕ(x, x  ) ∈ m ∞ n ϕ(x, x  ) = k  i=1 ∂f(x, x  ) ∂x i ξ i (x, x  ) + n  j=k+1 ∂f(x, x  ) ∂x j η j (x, x  ), (3.6) ξ i (x, x  ) η j (x, x  ) ε(n) ξ i (x, x  ) ∈ m ∞ n , i = 1, 2, . . . , k η j (x, x  ) ∈ m ∞ n , j = k + 1, . . . , n. m ∞ n ⊂< ∂f ∂x k+1 , , ∂f ∂x n > ε(n) + x 1 x k .ε(n). (3.7) ξ(x, x  ) ∈ m ∞ n g(x, x  ) = ξ(x, x  ) − ξ(0, x  ). g(x, x  ) ∈ m ∞ n g(0, x  ) = 0 g(x, x  ) = x 1 x 2 x k Q(x, x  ) ∈ x 1 x k .ε(n). ξ(x, x  ) = ξ(0, x  ) + x 1 x 2 x k Q(x, x  ). (3.8) ξ(0, x  ) ∈ ε(n−k) ξ(0, x  ) = n  i=k+1 ∂f(0, x  ) ∂x i η i (0, x  ), (3.9) f(0, x  ) ∈< f(x, x  ) > ε(n) ξ(0, x  ) ∈< ∂f ∂x k+1 , , ∂f ∂x n > ε(n) + x 1 x k .ε(n). (3.10) ξ(x, x  ) ∈< ∂f ∂x k+1 , , ∂f ∂x n > ε(n) + x 1 x k .ε(n) ϕ(x, x  ) ∈ m ∞ n ξ i (x, x  ) ξ i (x, x  ) = n  l=k+1 ∂f ∂x l h l (x, x  ) + x 1 x k .h(x, x  ), i = 1, 2, . . . , k. (3.11) ϕ(x, x  ) = k  i=1 ∂f ∂x i  n  l=k+1 ∂f ∂x l h l (x, x  ) + x 1 x k .h(x, x  )  + n  j=k+1 ∂f ∂x j ξ j = k  i=1 x i ∂f ∂x i  k  j=1,j=i x j  h(x, x  ) + n  j=k+1 ∂f ∂x j  h j (x, x  ) k  i=1 ∂f ∂x i + ξ j  . ϕ(x, x  ) ∈< x 1 ∂f ∂x 1 , , x r ∂f ∂x r , ∂f ∂x r+1 , , ∂f ∂x n > ε(n). ⇒ g(x 1 , , x n ) = (x 2 1 + +x 2 n ) 2 g ∈ m 4 n g ∈ m ∞ n g = k  i=1 x i ∂f ∂x i ξ i + n  i=k+1 ∂f ∂x i η i ≤  k  i=1  x i ∂f ∂x i  2 + n  i=k+1  ∂f ∂x i  2  1 2  k  i=1 ξ 2 i + n  i=k+1 η 2 i  1 2 ≤ c  k  i=1  x i ∂f ∂x i  2 + n  i=k+1  ∂f ∂x i  2  1 2 , c  k  i=1 ξ 2 i + n  i=k+1 η 2 i  1 2  x ≤ δ δ > 0  k  i=1  x i ∂f ∂x i  2 + n  i=k+1  ∂f ∂x i  2  1 2 ≥ 1 c (x 2 1 + + x 2 n ) 2 ⇔  k  i=1  x i ∂f ∂x i  2 + n  i=k+1  ∂f ∂x i  2  1 2 ≥ 1 c  x  4 ⇔  k  i=1  x i ∂f ∂x i  2 + n  i=k+1  ∂f ∂x i  2  ≥ 1 c 2  x  8 . C = 1 c 2 , α = 8 ⇒ g ∈ ε(n) ϕ = g − f ϕ ∈ m ∞ n f t (x) = f(x) + tϕ(x), t ∈ [0, 1] C, α, δ ≥ 0 r  i=1  x i ∂f t ∂x i  2 + n  i=r+1  ∂f t ∂x i  2 ≥ C  x  α ,  x < δ t ∈ [0, 1] r  i=1  x i ∂f t ∂x i  2 + n  i=r+1  ∂f t ∂x i  2 = G −ϕ = G 2 (−ϕ) G 2 =  k  i=1  x i ∂f t ∂x i  2 + n  i=k+1  ∂f t ∂x i  2  2 −ϕ G 2 . ξ =  r  i=1  x i ∂f t ∂x i  2 + n  i=r+1  ∂f t ∂x i  2  −ϕ G 2 , −ϕ =  r  i=1  x i ∂f t ∂x i  2 + n  i=r+1  ∂f t ∂x i  2  ξ, ξ x t η(x, t) = (η 1 , , η n ) η i (x, t) = x 2 i  ∂f t ∂x i  ξ i , i = 1, 2, , r η j (x, t) =  ∂f t ∂x i  ξ j , j = r + 1, , n. −ϕ = ∂f t ∂x η(x, t) η i x t  ∂h ∂t = η(h(x, t), t), h(x, 0) = x, h(x, t) η(0, t) = 0 h(0, t) = 0 h(x, t) : (IR n , 0) → (IR n , 0) t h ∈ Dif f r (n) f t (x) = f (x) + tϕ(x) ∂f t ∂t = ϕ(x) ∂ ∂t (f t ◦ h) = ∂f t ∂x ◦ h ∂h ∂t + ∂f t ∂t ◦ h = ∂f t ∂x ◦ h.η(h(x, t)) + ∂f t ∂t ◦ h = −ϕ(h(x, t)) + ∂f t ∂t ◦ h(x, t) = 0. f t ◦ h t f ◦ h(x, 0) = f(x) = f 1 ◦ h(x, 1) = g(h(x, 1)). f ω Dif f r (n)  3.2. Remark. (1) r = 0 (2) r = 1 f (3) r = 1 Remark. ω Diff r (n) ω K| x 1 , ,x r K| x 1 , ,x r {x 1 = 0}, . . . , {x r = 0}. Acknowledgement References [1] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenca, , Birkh¨auser, Boston-Base l-Stuttgart, 1985. [2] N. T. Cuong, N. H. Duc, N. S. Minh, H. H. Vui, , C. R. Aced. Sc. Paris. 285 (1977). [3] Michael Demazure, , Printed in Germany, Springer-Verlag, Berlin Heidelberg 2000. [4] Nguyen Tien Dai, Nguyen Huu Duc et Fr´ed´eric Pham, `e `e `e , Memoire de la societe Mathematique de France, Nouvelle s´erie N o 6, Suppl´ement au Bulletin de la Soci´et´e Mat´ematique de France, (3) 109 (1981). [5] John N. Mather, C ∞ , Publica- tions math´ematiques de l ´ I.H. ´ E.S., 35 (1968), 127-156. [6] John N. Mather, C ∞ , Ann. Math., 89 (1969), 254-291. [7] D. Siersma, , Quart. J. Math. Oxford (2) 32 (1981), 119 - 127. [8] H. H. Vui, , Thesis of doctorate, 1980. ω r ω f ∈ ε(n) Diff r (n) r f ∈ ε(n) ω Diff r (n) . 0}, . . . , {x r = 0}. Acknowledgement References [1] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenca, , Birkh¨auser, Boston-Base l-Stuttgart, 1985. [2] N. T. Cuong, N. H. Duc, N. S. Minh, H (3) 109 (1981). [5] John N. Mather, C ∞ , Publica- tions math´ematiques de l ´ I.H. ´ E.S., 35 (1968), 12 7-1 56. [6] John N. Mather, C ∞ , Ann. Math., 89 (1969), 25 4-2 91. [7] D. Siersma, , Quart Michael Demazure, , Printed in Germany, Springer-Verlag, Berlin Heidelberg 2000. [4] Nguyen Tien Dai, Nguyen Huu Duc et Fr´ed´eric Pham, `e `e `e , Memoire de la societe Mathematique de France, Nouvelle s´erie