ii

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1 213/ ,,, Pfaffian 1,, Borel,,, rigidity,,, Pfaff,, JSPS ( ). i

2 ii

3 x t n 2 1 e 1 2 dt HGM (holonomic gradient method) 1 1 holonomic 1.1 Hilberlt Hilbert F k F k = {(x, y) N 2 x + y k} 1 NHK 1

4 ( 1 ) k k p k k Figure 1: Hilbert F k ( ) 2+k (k +2)(k +1) #F k = = 2 2 = 1 2! k2 + O(k). O(k m ) k O(km ) k m k F k \(p + N 2 ). 1 # ( F k \(p + N 2 ) ) ( ) ( ) 2+k 2+k p = 2 2 (k +2)(k +1) (k p +2)(k p +1) = 2! 2! = Ck + O(1) = O(k) p = p 1 + p 2,p=(p 1,p 2 ). 2

5 ( ) m # F k \ (P (i) + N 2 ) k=1 2 k k Figure 2: Hilbert 2 (k ) k k K Q, C B: R( ) K[x 1,x 2 ] I x p 1 1 x p 2 2 F k F k F k F k F k = { α k c α x α 1 1 x α 2 2 c α K} 2 3

6 ( ) α 1 + α 2 F k K- 1. I = x p (x p = x p 1 1 x p 2 2 ) F k /(F k I) K ( ) ( ) k +2 k p +2 dim K F k /(F k I) = k 2 Proof. x α = x α 1 1 x α 2 2 I α p + N 2 xα F k I α p + N 2 F k β F k \(p + N 2 ) x β F k /(F k I) K- base I = x p(1),...,x p(m) 2. dim K F k /(F k I) k k K[x 1,x 2 ] I = x 1 x 2,x 2 1 F k /(F k I) base 3 k Figure 3: I = x 1 x 2,x 2 1 1,x 1,x 2,x 2 2,...,x k 2 4

7 I Hilbert H(k; I) H(k; I) =O(k d ) d I Krull (Krull d V (I) V (I) manifold d manifold ) D n = K x 1,...,x n, 1,..., n i x j = x j i + δ ij x i x j = x j x i i j = j i K K derivation 3 multi-index x α β = x α 1 1 x αn n 1 1 n n x α β >x α β α + β > α + β or ( α + β = α + β (α, β) (α, β ) ) Graded lexicographic order l = c αβ x α β +(< ) K[x, ξ] in < (l) =c αβ x α ξ β K[x, ξ] < l initial D n I ( < ) in < (I) = in < (l) l I K[x, ξ] in < (I) I Hilbert H(k; I) =H(k;in < (I))

8 Hilbert Hilbert Hilbert computer algebra hilbert... ( ) 3. I = 1,..., n in < (I) = ξ 1,...,ξ n ( ) K[x, ξ]/in < (I) = K[x] hilbert H(k; I) = ( ) k+n n = O(k n ) n F k = { α + β k c αβx α β } 4. k dim K F k /(F k I) =H(k; I) =H(k;in < (I)) ([2] 1.6, 6.6 ) I in < (I) 5 (Bernstein ). D n D n I H(k; I) =O(k m ) m n Krull Krull n 3 Krull n lower bound D n Krull 6

9 n 1 n 2 Bernstein [2] 6.8 m = n I holonomic ideal M = D n /I holonomic D n -module 45 B: I holonomic... initial initial Hilbert B: generator holonomic Macaulay2 is.holonomic Risa/asir ns twistedlog.holonomic 4 B: initial ideal generator initial Holonomic holonomic C: 4 import("ns twistedlog.rr");. 7

10 involutive base C:... blacket main theorem [4] [2] 6 HGM Björk [3] 6. D n /I holonomic D n -module D n /(I + n D n ) holonomic D n 1 -module M = D n /I,M k = F k /(F k I) I module M/ n M = n, x = x n : M M M k 1 M k ( m ) ( m ) dim k M k /( M M k ) n! kn + O(k n 1 ) n! (k 1)n + O(k n 1 ) = O(k k 1 ) K F k /(F k (I + D)) = M k /( M) M k OK. : M M ( ) N D n -module ( ) N M N = {m M k k m = inm} M = M/N D n -module : M M D n -module M/ M = M/ M 8

11 ( ). N D n n -module m N x n m N k m = k+1 (x n m)=x n k+1 m +(k +1) k m = ( ). m N k m = k k x k = x k k 1 + i! (k(k 1) (k i +1))2 x k i k i k x k m = x k k + i=1 x i i =( x 1)( x 2) ( x i) k i=1 = (...)m + k! 1 i! (k(k 1) (k i +1))2 x k i k i m k i=1 ( 1) k i k! i!(k i)! m m = (...)m M D n /(I + n D n ) D n 1 module k n m m>m x m n m l i x i n, i= l i D n 1 1,x n,x 2 n,...,x m n V k = { α n β n k c αβx α β } kashiwara-malgrange filtration M holonomic b(s) K[s] l V 1 s.t. b( n x n )+l I ( [6, Theorem 5.1.2, 5.1.3]) b(s) b x m n (b( n x n )+l) =b( n x n m)x m n + x m n l = n (...)x m n + b( m)x m n + x m n l b( s) = m m mod I + n D n x m n x n m 9

12 2 HGM, χ 2 ( ) holonomic gradient method χ 2 B: ( HGM ) hypergeometric module holonomic gradient method. 2.1 Γ Γ T n (x) { x n 2 1 e x 2 /N T (n) x> T n (x) = x N T (n) 1 ( N T (n) = x n 2 1 e x n n ) 2 =2 2 Γ 2 T n (x) T n(x)dx = 1 T n (x) T n (x) χ 2 χ 2 e (x m)2 2σ 2 /N, N = 2πσ 2 m σ 2 N(m, σ 2 ) χ 2 2 1

13 7. X 1,...,X n N(, 1) X 1 n Y = Xi 2 =(X 1,...,X n ) i=1 X n n χ 2 (random variable) (Ω, F,P) (R, B) 1 Proof. n ([8] 7.5 (p.171) ) n =1 Y = X 2 c P (Y <c)=p ( c<x< c) X N(, 1) P (a X b) = b y = x 2 P (Y <c)= = c c c a 1 2π e 1 2 x2 dx (a b) 1 c e x2 dx =2 e 1 2 x2 dx 2π 2π 1 2π y 1 2 e 1 2 y dy = c T 1 (y)dy Y 1 χ 2 n 8. X,Y X Y f(x),g(y) X+Y f g(x) = f(x t)g(t)dt 11

14 T n T 1 T n+1 (T n T 1 )(x) = 1 N T (n)n T (1) e x 2 x (x t) n 2 1 t 1 2 dt (T n T 1 ) t = xs x n (1 s) n 2 1 s 1 n 1 2 dt = x 2 Γ( n)γ( 1) 2 2 Γ ( ) n+1 2 R d matrix hypergeometric 1 F 1 zonal ( ) 1 2 A A, (1, )

15 8 P ({ }) =P ({ }) = 3 8 P ({ }) =P ({ }) = 1 8 ( ) P = 1 4 =25% P 5% 1% P 25% : A N(,.25) rk() 7 7, n k=1 rk()2 /(.25) 2 n χ n rd() 2 y 2. y (.25) 2 =ȳ 3. ȳ T ȳ n (x) 5% 9. ( (.25) 2 ) rd() n =5 P 5% P 5% n ( n =5) R P > N< 5; > a< runif(n).5; > b< sum(aˆ2)/ (.25)ˆ2; > p< 1 pchisq(b,df=n ) ; > p Listing 1: p-value 13

16 [1] # [1] # [1] #. > curve( pchisq(x,df=5),from=,to=1); 7 D.Knuth,, pchisq(x, df = 5) x Figure 4: χ 2 pchisq, x T n(t)dt 2.2 HGM P γ(x) = x t n 2 1 e t 2 dt 1 γ(ȳ)/γ(+ ) P - 14

17 [,x] γ(x) Heaviside H(x t)h(t)t n 2 1 e t 2 dt Heaviside { 1 t H(t) = t< γ(x) ( ) t H(t) =δ(t) tδ(t) = δ(t) delta t t H(t) = f(t, x) =H(x t)h(t)t n 2 1 e t 2, t f(t, x) =H (t)h(x t)g(t) H(t)H (x t)g(t)+h(t)h(x t)g (t) delta (x t)t (x t)t t f(t, x) =H(t)H(x t)g (t) g (t) g(t) ( ( n ) l 1 := (x t) t t ) 2 t l 2 := (x t) x f(t, x) D = K x, t, x, t I = Dl 1 + Dl 2 I f ( l I l f = ) 15

18 (I + t D) K x, x l l l = P 1 + t P 2 (P 1 I,P 2 D) (1) f(x, t)dt = P 1 f(x, t)dt + =[P 2 f] = t P 2 f(x, t)dt l γ(x) = (O. D. E. ) ( ) 1. D/I Proof. x >t> x > t graded lexicographic order. in < (l 1 )=2xtξ t,in < (l 2 )= xξ x ξ t =in < ( t ), ξ x =in < ( x ). xtξ t,xξ x Hilbert 3 ( ), x l 1 (2t t )l 2 (t n +2)l 2, 2t x +2t t + t n +2, xtξ t,xξ x,tξ x Hilbert 2 ( ) I I = D, Bernstein D/I K x, x /(I + t D) K x, x 6, ( [2] ) (I + t D) K x, x ( ) (1) l γ(x) Risa/Asir ( Risa/Asir ) J = 2x 2 x +(x n +2) x Listing 2: import( nk restriction. rr ); L=[(x t) (t dt (n/2 1)+t /2), (x t) dx ] ; G= n k restriction. integration ideal(l,[t,x],[dt,dx],[1,]); holonomic gradient method (HGM) γ(x) N T (n) =γ(+ ) 16

19 Risa/Asir Risa/Asir C C Risa/Asir C [1] HGM H. G. M.. Step 1. holonomic J Step 2. J Pfaffian Step 3. Step 2 Pfaffian HGM Step 2 Pfaffian Q = P i Q, i =1,...,n x i ( [2] 6.2 ) n =1 Holonomic Pfaffian Pfaffian grad(q) =(P 1 Q,..., P n Q) Q holonomic gradient method γ(x) Step 1 Step 2 (2x x 2 +(x n +2) x ) γ(x) = ) ( )( ) 1 γ(x) x ( γ(x) x γ(x) = 1 (x n +2) 2x x γ(x) 17

20 Step 3. γ(x) x t n 2 1 e t 2 dt = = = k= k= k= x 1 k! 1 k! ( t 2) k dt t n k! ( 1 ) k x t n 2 1+k dt 2 ) k x n 2 +k ( 1 2 n + k 2 = n 2 x n2 1F 1 ( n 2 ; n 2 +1; x 2 x n γ(x) =e x 2 x n 2 h(x) h(x) θ ( ) θ + n 2 x(θ +1) 2 Pfaffian γ(x)/n T (N) HGM Risa/Asir Listing 3: p-value import( names. rr )$ import( taka runge kutta. rr )$ Glib math coordinate=1$ extern Aig$ / Df : N : approx degree / def poch(a,n) { R=1; for (I=; I<N; I++) { R=R (A+I ) ; } return R; } def igs (X, Df,N) { S = 1; for (K=1; K<=N ; K++) { S += eval ((X 1/2)ˆK)/poch(Df/2+1,K); } ) 18

21 return S/(Df/2); } def igs (X, Df,N) { return eval( igs (X, Df,N) exp ()); } def igs (X,Df,N) { return igs (X, Df,N) eval(xˆ(df/2) exp( X/2)); } / diff(igs), gamma(x). / def igs1 (X, Df,N) { S = ; for (K=1; K<=N ; K++) { S += (1/2) K eval ((X 1/2)ˆ(K 1))/poch(Df/2+1,K); } return S/(Df/2); } def igs1 (X, Df,N) { return eval( igs1 (X, Df,N) exp ()); } def ig (X, Df) { / h(x) / Step=.1; X=1; N=4; / / Iv=[igs (X,Df,N), igs1 (X,Df,N)]; Eq = [ [, 1 ], [ 1 / ( 2 x),( 1/x) (Df/2+1 x/2)]]; / Runge Kutta / A= t k rk. runge kutta 4 linear(eq,x,[],x,iv,x,step); return A; } / Df gamma(x). / def ig(x,df) { extern Aig ; Aig=ig (X, Df ); A=[]; for (I=; I<length(Aig); I++) { V=A i g [ I ] ; X=V [ ] ; A= cons([x,v[1] eval(exp( X/2) Xˆ(Df / 2 ) ) ],A) ; } return reverse(a); } / N T(Df) / def nc(df) { return(pari(gamma,df/2) eval (2ˆ(Df/2))); } end$ γ(7) γ(7)/n T (5) [1893] load("evalig3.rr"); [198] A=ig(7,5)$ [1981] A[]; [7, E31] [1982] A[][1]/nc(5);

22 χ 2 HGM χ 2 Numerical Recipes HGM [11] HGM [1] F k = {(m 1,m 2 ) N 2 m 1 + m 2 k}. p, q N 2, F k \ (p + N 2 ) (q + N 2 ) k (k ). n? 3.2. [1] Q x,. 1. x k k = θ(θ 1) (θ k +1), θ = x. 2. b(θ)x k = x k b(θ + k). b(θ) θ = x. 3. k x k = x k k + k 1 i=1 (k(k 1) (k i i! +1))2 x k i k i [1] e xt tn t [15] 7 T 5 (t)dt [2] 1. 1 F 1 (a, c; x). 2. (a, c ) [45] ( ) χ 2 1 F 1. 1 F 1 zonal, positive definite symmetric matrix.. : 2

23 1. A.G.Constantine, Some Non-Central Distribution Problems in Multivariate Analysis, The Annals of Mathematical Statistics 34 (1963), H.Hashiguchi, Y.Numata, N.Takayama, A.Takemura, The holonomic gradient method for the distribution function of the largest root of a Wishart matrix, Journal of Multivariate Analysis, 117, (213) References [1] D.Cox, J.Little, D.O Shea, Ideals, Varieties, and Algorithms, Springer.. 1 1,. [2] JST CREST,,. [3] Björk, Rings of Differential Operators. Weyl. [4], D,. [5] T.Kimura, Hypergeometric Functions of Two Variables... [6] M.Saito, B.Sturmfels, N.Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Springer. [7],, 2,. R. [8],,.. [9] [1] [11] 21

24 22

25 Pfaffian 1 ( ) Sep. 2,3, 213, Introduction Gauss F (a, b, c; x) = n= (a, n)(b, n) (c, n)(1,n) xn, (a, n) =Γ (a + n)/γ (a), c, 1, 2,... {x C x < 1}., Γ (c) t b c (t x) b (t 1) c a dt Γ (a)γ (c a) t 1 1 (Re(c) > Re(a) > )., [, 1], t 1/t., [ x(1 x)( d dx )2 +{c (a+b+1)x}( d ] dx ) ab f = (1)., Pfaffian df = Ωf, f G = ( F (a, b, c; x) f = d F (a, b, c; x) dx ) ( 1, Ω = ab x(1 x) c (a+b+1)x x(1 x) ( ) 1 x 1 ϕ = Gf ϕ Pfaffian b dϕ = dg f + G df = dg G 1 ϕ + GΩG 1 ϕ [( ) ( dx b = a c x + c a b 23 ) dx x 1 ) ] ϕ dx.

26 ., G d log x = dx dx, d log(x 1) = x x 1, 214. X = C {, 1} x U x, (1) Sol(U x ) 2. ẋ (, 1). ẋ X γ, Sol(Uẋ) f. γ γ X homotopic, f. ρ ẋ loop, ρ M ρ (f) f, Sol(Uẋ) M ρ (c 1 f 1 + c 2 f 2 )=c 1 M ρ (f 1 )+c 2 M ρ (f 2 ) (f 1,f 2 Sol(Uẋ), c 1,c 2 C), M ρ Sol(Uẋ). loopρ loop ρ loop ρ ρ M ρ ρ(f) =M ρ (M ρ (f)). π 1 (X, ẋ) ρ M ρ GL(Sol(Uẋ)),. (1). X ẋ loop ρ 1 loop ρ 1. M = M ρ M 1 = M ρ1 Sol(Uẋ). Sol(Uẋ) M M 1 M M 1. M M 1., M M 1. twisted homology group., twisted homology group,, F 1.,,,,. 1 Stokes F (a, b, c; x) C x = C {, 1,x} 1 u(t)ϕ(t) u(t) = t b c (t x) b (t 1) c a =(t x ) α (t x 1 ) α 1 (t x 2 ) α 2, { dt ϕ(t) = t 1, x =, x 1 = x, x 2 =1, x 3 =, α = b c, α 1 = b, α 2 = c a, α 3 = a,. α j / Z, x t. C x k k ψ u(t) u(t)ψ. u(t) ψ ψ 24

27 u(t) u pairing ψ, u. Stokes d(u(t)ψ) = u(t)ψ. D d(u(t)ψ) =du(t) ψ + u(t)dψ = u(t)(ω ψ + dψ), ( b c ω = d log(u(t)) = t D + b t x + c a ) dt = t 1 2 i= α i dt t x i,, ω ψ,d u, ω = d + ω, C x 1-form ω,.,, ψ, ( D) u. ω ω (D u )=( D) u, D u(t) D u(t) D u(t) D. Stokes, ω, ω. Theorem 1 ( Stokes ) ω ψ,d u = ψ, ω (D u ). ω ψ = C x C k ψ k, k D j u(t) γ = j J Du j k, Ck u(c x). ω (γ) = k. ϕ = dt,. t 1 dt dt = C x. (1, ) u(t) (1, ) u,. 1, C x, (1, ). x (, 1), ϕ, γ u = 1 u(t)ϕ γ u. ε R, I 1+ε,R 1+ε R, C ε ε, C R R. γ u = I1+ε,R u 1 C1 u + 1 C 1 λ 2 1 λ, u λ i = e 2π 1αi 3 u(t), t, t 1, t x. Cauchy ϕ, γ u ε, R. Re(c) > Re(a) > lim u(t)ϕ =lim u(t)ϕ = ε C R 1 C ϕ, γ u = 1 u(t)ϕ. 25

28 C C 1 x 1 1+ε I 1+ε,R R ω (γ u ). ω (I u 1+ε,R )=Ru(R) (1 + ε) u(ε+1). C 1, t 1 2π, ω (C u 1 )=λ 2 (1 + ε) u(1+ε) (1 + ε) u(1+ε). C, t, t x, t 1 2π, ω (C u )=λ 3 R u(r) R u(r)., ω (γ u )=. γ u. Problem 1 x (, 1), C x x =, x 1 = x, x 2 =1, x 3 = γ ij ( i<j 3) γ u ij. 2 γ u ε,r, ϕ, γ u.,. ϕ = dt t 1 Du C2 k (C x ) ω (D u ) pairing ϕ, ω (D u ), Theorem 1 ϕ, ω (D u ) = ω ϕ,d u =. C u 2 (C x ) w, ω H 1 (C x, ω ) : H 1 (C x, ω )=ker( ω : C u 1 (C x ) C u (C x ))/ ω (C u 2 (C x )). 26

29 Theorem 2 ( - ) dim C H 1 (C x, ω )=2. H 1 (C x, ω ) x, (1) Sol(U x ). u(t, x) X = {(t, x) P 1 X t(t x)(t 1) } P 1 P 1, C x ı x : C x pr 1 (x), pr : X(t, x) x X. H 1 ( ω )= x X H 1 (C x, ω ) H 1 ( ω,u x )= x U x H 1 (C x, ω ) Sol(U x ). H 1 (C x, ω ) Sol(U x ) x germ Sol(x). ϕ pairing, x. u(t) 1/u(t), H 1 (C x, ω ), H 1 ( ω )= x X H 1 (C x, ω ). d log(1/u(t)) = ω. H 1 (C x, ω ) H 1 (C x, ω ) I h. 1-chains + p i transversely. ( u + u 1 )= ( + ) pi u(p i )u 1 (p i ) i, I h. ( + ) pi γ + γ.,. p i I h (γ u, γ u 1 + )= I h (γ u +, γ u 1 ), ; α i = α i, λ i =1/λ i, u = u 1. Theorem 3 i<j 3, p<q 3, I h (γij, u (γpq) u )= 1 λ i λ j (1 λ i )(1 λ j ) if (i, j) =(p, q), λ i 1 λ i λ j if i = p, j > q, 1 λ j if j = p, 1 1 λ j if i < p, j = q, 1 if i<p<j<q, if i<p<q<j, i<j<p<q. 27

30 γ u = γ23, u γ1 u = γ12 u (γu ) =(γ23) u,(γ1 u ) =(γ12) u ) Proof. H = ( 1 λ2 λ 3 (1 λ 2 )(1 λ 3 ) 1 1 λ 2 λ 2 1 λ 2 1 λ 1 λ 2 (1 λ 1 )(1 λ 2 ). C u u 1 p 1 C 1 p 2 p 3 I 1+ε,R γ u (γu ). p 1 1, u u(p 1 )u 1 (p 1 )= λ 3, C 1 1 λ 3 λ 3 λ 3 1 λ 3. p 2 1, u u 1 u(p 2 )u 1 (p 2 )=1, C λ 2 ( 1) 1 1 λ 2. I h (γ u, γ u 1 ), 1 λ 2λ 3 (1 λ 2 )(1 λ 3 ). γ u (γu 1 ). p 3 1. u p 3 u u 1 u(p 3 )u 1 (p 3 )=1. C λ 2.. Problem 2. Remark 1 α j / Z (j =, 1, 2, 3). det(h) = 1 λ 1 λ 2 λ 3 (1 λ 1 )(1 λ 2 )(1 λ 3 ). 28

31 3 ẋ (, 1) ( X = C {, 1}), π 1 (X, ẋ) loop ρ. (1) Sol(Uẋ) ρ M ρ, (γ u, γ1 u ) M ρ M ρ, i.e., (M ρ (γ u ), M ρ (γ1 u )) = (γ u, γ1 u )M ρ,. M ρ, M ρ, a, b, c M ρ, Mρ, (γu, γ1 u ), (γ u, γ1 u ) H = ( I h (γi u, (γj u ) ) ). i,j Theorem 4. (1) I h (M ρ (γ+), u M ρ (γ u 1 )) = I h (γ+, u γ u 1 ), γ± u±1 H 1 (C x, ±ω ). (2) t M ρ HM ρ = H. (3) M ρ β, γ u I h (γ u, (γ u ) ) β β =1. (4) M ρ β 1, β 2 γ u 1, γ u 2. β 1 β 2 1 I h (γ u 1, (γ u 2 ) )=. Proof. (1) ẋ Uẋ, γ+, u γ u,.,. ( ) (2) γ u (γ u, γ1 u ) (γ u, γ1 u g ) (g,g 1 C), g 1 ( ) ( ) M ρ (γ u )=(γ u, γ1 u g g )M ρ, I g h (γ u, (γ u ) )=(g,g 1 )H 1 g1. (1) I h (M ρ (γ u ), (M ρ (γ u )) )=(g,g 1 ) t M ρ HM ρ (g,g 1 ) t M ρ HM ρ, g,g 1 t M ρ HM ρ = H. (3) M ρ (γ u )=βγ u, (1) ( g g 1 ) =(g,g 1 )H ( g g 1 ( g g 1 ) ). I h (γ u, (γ u ) ) = I h (M ρ (γ u ), M ρ ((γ u ) )) = I h (βγ u, β (γ u ) ) = (β β ) I h (γ u, (γ u ) ). I h (γ u, (γ u ) ) β β =1. 29

32 (4) (1) I h (γ u 1, (γ u 2 ) ) = I h (M ρ (γ u 1 ), M ρ ((γ u 2 ) )) = I h (β 1 γ u 1, β 2 (γ u 2 ) ) = (β 1 β 2 ) I h (γ u 1, (γ u 2 ) ). β 1 β 2 1 I h (γ u 1, (γ u 2 ) )=. Remark 2 Theorem 4 (4), I h (γ2 u, (γ2 u ) ) β2 β 1 β2 1 β 1 β 2. =1/β 2, Lemma 1 M 1 λ λ 1 = e 2π 1c. M 1 γ u 23, M λ λ 1 γ u 1. λ λ 1 1 I h (γ u 1, (γ u 23) )= γ u 1, γ u 23. Proof. (1, ) ρ. u(t) = t α (t x) α 1 (t 1) α 2 x =ẋe 2π 1θ (θ [, 1]). (t x) α 1. x, t (1, ) t arg(t x) ρ. M 1 (γ23) u =γ23. u γ1 u ρ. t = t = x =ẋe 2π 1θ (θ [, 1]) γ 1 (, 1) γ 1 : t =ẋe 2π 1θ s s (, 1). γ 1 ρ, γ 1. u(t) (, 1), ρ. γ 1(u(t)) = (ẋe 2π 1θ s) α (ẋe 2π 1θ s ẋe 2π 1θ ) α 1 (ẋe 2π 1θ s 1) α 2 = e 2π 1θ(α +α 1)ẋ α +α 1 s α (s 1) α 1 (ẋe 2π 1θ s 1) α 2. ρ θ 1 u(t) λ λ 1, M 1 (γ1) u =λ λ 1 γ1. u λ λ 1 1 Theorem 4, I h (γ1, u (γ23) u )=. Theorem3, I h (γ1, u (γ1) u ) I h (γ23, u (γ23) u ),. Lemma 2 M 1 1 λ 1 λ 2 = e 2π 1(c a b). M 1 1 γ u 3, M 1 λ 1 λ 2 γ u 12. λ 1 λ 2 1 I h (γ u 3, (γ u 12) )= γ u 3, γ u 12. Problem 3 Lemma 2 Lemma 1. 3

33 Theorem 5 ( ) M, M 1, : M (γ u ) = γ u (1 λ )(1 λ 1 )I h (γ u, (γ1) u )γ1 u = λ λ 1 [γ u (1 λ 2 )(1 λ 3 )I h (γ u, (γ23) u )γ23] u, M 1 (γ u ) = γ u (1 λ 1 )(1 λ 2 )I h (γ u, (γ12) u )γ12 u = λ 1 λ 2 [γ u (1 λ )(1 λ 3 )I h (γ u, (γ3) u )γ3] u. Proof. λ λ 1 1, M 1, λ λ 1 γ23, u γ1 u. M (γ1) u = γ1 u (1 λ )(1 λ 1 )I h (γ1, u (γ1) u )γ1 u = γ1 u (1 λ )(1 λ 1 )(1 λ λ 1 ) γ1 u = λ λ 1 γ u (1 λ )(1 λ 1 ) 1, M (γ23) u = γ23 u (1 λ )(1 λ 1 )I h (γ23, u (γ1) u )γ1 u = γ23. u. λ λ 1 =1. Remark 3 λ λ 1 1, M (γ u )=γ u (1 λ λ 1 )I h (γ u, (γ u 1) )I h (γ u 1, (γ u 1) ) 1 γ u 1 γ 1 λ λ 1 I h. Corollary 1 H 1 (C x, ω ) (γ u, γ u 1 )=(γ u 23, γ u 12) M, M 1 M,M 1 M = I 2 (1 λ )(1 λ 1 )r t 1 r1 t H ( ) 1 = λ λ 1 [I 2 (1 λ 2 )(1 λ 3 )e t e t λ λ H]= 1 λ 2 1, λ λ 1 M 1 = I 2 (1 λ 1 )(1 λ 2 )e t 1 e t 1 H ( 1 = λ 1 λ 2 [I 2 (1 λ )(1 λ 3 )r t 3 r3 t H]= 1 λ 1 λ 1 λ 2 I 2, e = t (1, ), e 1 = t (, 1), r 1 = 1 ( ) 1 λ λ 1 λ 2, r 1 λ 1 λ λ 3 = λ ( 1 λ1 λ λ 1 λ 1 Proof., γ u i (γ u, γ u 1 ), γ u = g γ u + g 1 γ u 1 H 1 (C x, ω ) ). ), ( g1 e i (i =, 1). γ u 1 H 1 (C x, ω ) γ u, γ u 1 g 2 ) 31

34 g γ u + g 1 γ1 u. Theorem 3 = I h (γ1, u (γ23) u ) = g I h (γ23, u (γ23) u )+g 1 I h (γ12, u (γ23) u ) = H g + H 1 g 1, λ 1 1 λ 1 = I h (γ1, u (γ12) u ) = g I h (γ23, u (γ12) u )+g 1 I h (γ12, u (γ12) u ) = H 1 g + H 11 g 1. ( ) ( ) λ 1 = t g H, γ1 1 λ 1 g u 1 ( ) r 1 = t H 1 λ 1 = 1 ( ) 1 λ λ 1 λ 2 1 λ 1 1 λ 1 λ λ 1 ( ). Theorem 5 M (γ u g1 ) M. γ u γ1 u ( g1 g 2 ) = I 2 ( g1 g 2, M (γ u ) ) g 2 r 1. I h (γ u, (γ u 1) )=(g 1,g 2 ) Hr 1 = t r 1 t H [ I2 (1 λ )(1 λ 1 )r 1 t r 1 t H ] ( g 1 g 2. e i = e i. M, M 1., λ λ 1 =1 λ 1 λ 2 =1. ( g1 g 2 ). ) 4 Appell s F 1 Appell s F 1 {x =(x 1,x 2 ) C 2 max( x 1, x 2 ) < 1} F 1 (a, b 1,b 2,c; x) = (a, n 1 + n 2 )(b 1,n 1 )(b 2,n 2 ) (c, n 1 + n 2 )(1,n 1 )(1,n 2 ) xn 1 1 x n 2 2 n N 2, c, 1, 2,. Γ (c) Γ (a)γ (c a) 1 t b 1+b 2 c (t x 1 ) b 1 (t x 2 ) b 2 (t 1) c a dt t 1 32

35 (Re(c) > Re(a) > ). x =,x 3 =1,x 4 =, t = x i exponent α i (i =,...,4) : x =, x 1, x 2, x 3 =1, x 4 = ; α = b 1 + b 2 c, α 1 = b 1, α 2 = b 2, α 3 = c a, α 4 = a. F 1 (a, b 1,b 2,c; x), : x 1 (1 x 1 ) 2 1 +x 2 (1 x 1 ) 1 2 +[c (a+b 1 +1)x 1 ] 1 b 1 x 2 2 ab 1, x 2 (1 x 2 ) 2 2 +x 1 (1 x 2 ) 1 2 +[c (a+b 2 +1)x 2 ] 2 b 2 x 1 1 ab 2, (x 1 x 2 ) 1 2 b 2 1 +b 1 2. F 1 (a, b 1,b 2,c) Appell s F 1. Fact 1 S = {x C 2 x 1 x 2 (x 1 1)(x 2 1)(x 1 x 2 )=}. X = C 2 S x X U x F 1 (a, b 1,b 2,c) Sol(U x ). X ẋ =(ẋ 1, ẋ 2 ) ẋ =< ẋ 1 < ẋ 2 < 1=ẋ 3. ρ ij ( i< j 3, (i, j) (, 3)) X ẋ loop x j x i ẋ j,,., i = i, j. Fact 2 π 1 (X, ẋ) ρ ij. 5 F 1 (a, b 1,b 2,c) C x = C {,x 1,x 2, 1}, u(t) =t α (t x 1 ) α 1 (t x 2 ) α 2 (t 1) α 3,, H 1 (C x, ω ), H 1 ( ω )= x X H 1 (C x, ω ), germ Sol(x) H 1 (C x, ω ). x i x j γ ij ( i<j 4) γ u ij. u(t) 1, H 1 (C x, ω ), H 1 ( ω )= x X H 1 (C x, ω ), H 1 (C x, ω ) H 1 (C x, ω ) I h. Theorem 3 i<j 4, p<q 4. λ i = e 2π 1α i ( i 4). 33

36 6 F 1 (a, b 1,b 2,c) π 1 (X, ẋ) ρ M ρ GL(Sol(ẋ)) F 1 (a, b 1,b 2,c). M ρij = M ij ( i<j 3, (i, j) (, 3)). Theorem 4,. Lemma 3 M ij 1, λ i λ j. 1, λ i λ j γ u ij. Proof. Lemma 1, λ i λ j γij. u {i, j, k, l, m} = {,...,4} k, l, m x k x l x k x m 1. Theorem 6 i<j 3, (i, j) (, 3) M ij (γ u )=γ u (1 λ i )(1 λ j )γ u iji h (γ u, (γ u ij) ). Proof. λ i λ j 1. Theorem 4 M ij 1 {γ u H 1 (C x, ω ) I h (γ u, (γ u ij) )=}. γ u I h (γ u, (γ u ij) )=, γ u, 1. γ u = γ ij, Theorem 3 γ u ij (1 λ i )(1 λ j )γ u iji h (γ u ij, (γ u ij) ) = γij u (1 λ i)(1 λ j )(1 λ i λ j ) γij u =(λ i λ j ) γij u (1 λ i )(1 λ j ), γ ij λ i λ j. Lemma 3, M ij. Problem 4 H 1 (C x, ω ), H =(I h (γ u i, (γ u i ) )) ij., M ij. Problem 5 ρ 14 π 1 (X, ẋ) x 1 R 1(R>>),, R, ẋ 1. M 14 = M(ρ 14 ).,, γ u 14. Problem 6 x C {} m simplex x = {t =(xs 1,..., xs m ) C m s 1,...,s m >,s s m < 1} u(t, x) =(x t 1 t m ) α t α 1 1 t α m m, x, e 2π 1(α +α 1 + +α m) u(t, x). Problem 7,. 34

37 References [Apk] Appell P. and Kampé de Fériet M. J., Fonctions hypergéométriques et hypersphériques: polynomes d Hermite, Gauthier-Villars, Paris, [AoK] Aomoto K. and Kita M., translated by Iohara K., Theory of Hypergeometric Functions, Springer Monographs in Mathematics, Springer Verlag, 211. [CM] Cho K. and Matsumoto K., Intersection theory for twisted cohomologies and twisted Riemann s period relations I, Nagoya Math. J., 139 (1995), [G] Goto Y., The monodromy representation of Lauricella s hypergeometric function F C, preprint, 214, [math.ag]. [GM] Goto Y. and Matsumoto K., The monodromy representation and twisted period relations for Appell s hypergeometric function F 4, to appear in Nagoya Math. J. [KN] Kita M. and Noumi M., On the structure of cohomology groups attached to the integral of certain many-valued analytic functions. Japan. J. Math., 9 (1983), [L] Lauricella G., Sulle funzioni ipergeometriche a più variabili, Rend. Circ. Mat. Palermo, 7 (1893), [M1] Matsumoto K., Monodromy and Pfaffian of Lauricella s F D in terms of the intersection forms of twisted (co)homology groups, Kyushu J. Math., 67 (213), [M2] Matsumoto K., Pfaffian of Lauricella s hypergeometric system F A, preprint, 213. [MY] Matsumoto K. and Yoshida M., Monodromy of Lauricella s hypergeometric F A - system, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (214), [Y] Yoshida M., Hypergeometric functions, my love, -Modular interpretations of configuration spaces-, Aspects of Mathematics E32., Vieweg & Sohn, Braunschweig,

38 36

39 Borel,, Introduction 213. Gevrey,,. : z k+1 d φ = A(z)φ. (.1) dz, k Z A(z) M(n; C{z}) C{z} n. A() Φ(λ) =det ( λ A() ) =. (.1)., k =, z =., Φ(λ) = λ 1,, λ n z =, (.1)., λ i λ j / Z (i j), (.1) P (z) GL(n; C{z}) φ = P (z)ψ : z d dz ψ = λ 1... λ n ψ. (.2) 37

40 diag(λ 1,, λ n ) Λ, (.1) P (z)z Λ., (.1) z = λ 1,, λ n. k 1., φ = P (z)ψ (.1) : z k+1 d dz ψ = Λ 1 (z)... ψ. (.3) Λ n (z), Λ j (z) (j =1,,n) Λ j () = λ j., P (z) GL(n; C[[z]]),, P (z). Λ j (z),,., Borel. Borel Laplace (Borel ) Laplace,,, Borel, (.1) P (z)exp( z z k 1 Λ(z)dz) ( Λ(z) =diag(λ1 (z),, Λ n (z)) ) (.1). 1 Gevrey (.1) P (z),, P (z) Poincaré rank,. Gevrey, Gevrey Gevrey. [Ba1], [Ra2] Gevrey. 38

41 1.1 Gevrey 1.1 (Gevrey ). ˆf(z) = f n z n C[[z]] Gevrey order k(> ), C>, n n= f n C n+1 Γ (1 + n/k) (1.1). Gevrey order k C[[z]] 1/k. k < k 1 C[[z]] 1/k1 C[[z]] 1/k,, k 1, C[[z]] 1/ := C{z} k> C{z} C[[z]] 1/k., m := z C[[z]] 1/k = { ˆf(z) C[[z]] 1/k f =} : 1.2. ( C[[z]] 1/k, m ) PID., f 1 ˆf(z) = j= ( 1) j (f(z) f ) j f j+1 ˆf(z) ˆf 1 (z), 1.3 1/ ˆf(z) C[[z]] 1/k, C[[z]] 1/k., PID, C[[z]] 1/k m C[[z]] f(z) C{z}, ĝ(z) m f(ĝ(z)) C[[z]] 1/k., ˆf(z) C[[z]] 1/k, d ˆf dz (z) = f n+1 (n +1)z n n=., d ˆf/dz C[[z]] 1/k C[[z]] 1/k C., C{z} C[[z]] 1/k, : 1.4. C[[z]] 1/k C{z}. 39

42 1.2 C S d, α, ρ (α,ρ > ) S = S(d, α, ρ) ={z = re iθ <r<ρ, θ d < α/2}., S(d, α) =S(d, α, ) ={z = re iθ θ d < α/2}., S = S(d, α, ρ) S 1 d 1, α 1, ρ 1 (α 1,ρ 1 > ) { } S 1 = S 1 (d 1, α 1, ρ 1 )= z = re iθ <r ρ 1, θ d 1 α 1 /2, S 1 S. S, f(z) S, ˆf(z) =., f(z), ˆf(z) N ν f (z,n). ν f (z,n) =z N (f(z) N 1 n= f n z n ) f n z n C[[z]] 1.5 ( ). f(z) S ˆf(z) Gevrey order k, S S 1, C>, N S 1 n= ν f (z,n) C N+1 Γ (1 + N/k) (1.2)., f(z) = k ˆf(z). A k (S), ˆf(z) C[[z]] S Gevrey order k., f(z) A k (S) ˆf(z) C[[z]]., T k : A k (S) C[[z]]. T k : A k (S) C[[z]] 1/k 4 C[[z]]

43 A k (S) C[[z]] 1/k T k., A () k (S) :=KerT k. : 1.6. A k (S), T k., A () k (S) A k(s)., k 2 >k 1 > A k2 (S) A k1 (S) k 2 >k 1 >., S k 1,k 2, A () k 1 (S) =A () k 2 (S)., f(z) A k1 (S), ˆf(z) C[[z]]1/k2 f(z) = k1 ˆf(z) f(z) Ak2 (S) f(z) A k (S) \ za k (S)., f(z) = k ˆf(z), f f(z )= z S 1/f(z) / A k (S). (S) f(z) A() k (S), S S 1, C> N> A () k f(z) C N+1 z N Γ (1 + N/k) S 1., f(z) A () k (S) : 1.7. f(z) A () k (S), S S 1, C, h > S Borel f(z) Ce h z k ˆf(z) C[[z]], Borel B k ( ˆf)(ζ) B k ( ˆf)(ζ) = n= f n Γ (1 + n/k) ζn. ˆf(z) C[[z]] 1/k B k ( ˆf)(ζ) C{ζ}., f(z) S(d, α, ρ) (α > π/k)., f(z) d Borel B k (f)(ζ) B k,d (f)(ζ) = 1 2πi γ 41 f(z)e (ζ/z)k z k d(z k ) (1.3)

44 ., γ :[, 1] S(d, α, ρ) ρ (, ρ), β (π/k, α) 3t ρ exp ( i(d + β/2) ) ( t 1/3) γ(t) = ρ exp ( i(d +3(1 2t)β/2) ) (1/3 t 2/3) 3(1 t) ρ exp ( i(d β/2) ) (1.4) (2/3 t 1)., B k,d (f)(ζ) S(d, α π/k) exponential size k,, S(d, α π/k) S 1 (d, β, ), C, h > B k,d (f)(ζ) Ce h ζ k s B k,d (z s )(ζ) = ζ s Γ (1 + s/k) (1.5)., Borel B k Borel B k C[[z]] : : B k ( ˆf)(ζ) = f n B k,d (z n ). n= 1.8. k>k 1 >, f(z) A k1 (S(d, α, ρ)) (α > π/k) f = k1 ˆf. k 1 2 = k 1 1 k 1, B k,d (f)(ζ) A k2 (S(d, α π/k)) B k,d (f)(ζ) = k2 Bk ( ˆf)(ζ)., k = k 1 B k,d (f)(ζ) C{ζ} B k,d (f)(ζ) = B k ( ˆf)(ζ). Proof. g(ζ) =B k,d (f)(ζ). ( ν g (ζ,n)=ζ N g(ζ) B ( N 1 )) k f n z n 42 n=

45 (1.5) ν g (ζ,n)=ζ N B k,d (z N ν f (z,n))(ζ)., ν f (z,n) (1.2), B k,d (1.3) ν g (ζ,n) : S(d, α π/k) S 1 C >, N ν g (ζ,n) C N+1 Γ(1 + N/k 1 )/Γ(1 + N/k)., B k,d (f)(ζ) = k2. Bk ( ˆf)(ζ)., k = k 1 B k ( ˆf)(ζ) 1.4. f(z) f (k) (z) =f(z 1/k ), B 1,kd B k,d B k,d (f)(ζ) =B 1,kd (f (k) )(ζ k )., ˆf(z) C[[z]] ˆf (k) (z) = ˆf(z 1/k ) C[[z 1/k ]], (1.5) B 1 C[[z 1/k ]] B k (f)(ζ) = B 1 (f (k) )(ζ k ). 1.4 Laplace α, ρ > g(ζ) A k1 (S(d, α, ρ)), g(ζ) = k1 ĝ(ζ). g(ζ) S(d, α) exponential size k. g(ζ) d Laplace L k,d (g)(z) L k,d (g)(z) =., ĝ(ζ) =. e id g(ζ)e (ζ/z)k z k dζ k (1.6) g n ζ n Laplace L k (ĝ)(z) n= L k (ĝ)(z) = g n Γ(1 + n/k)z n n= 43

46 1.5. s L k,d (ζ s )(z) =Γ (1 + s/k) z s (1.7)., Laplace L k Laplace L k C[[ζ]] : L k (ĝ)(z) = g n L k,d (ζ n ). : n= 1.9. k2 1 = k1 1 + k 1 ε > δ(ε) > L k,d (g)(z) A k2 (S(d, α + π/k ε, δ(ε))) L k,d (g)(z) = k2 Lk (ĝ)(z). Proof. S(d, α, ρ) g(ζ) = k1 ĝ(ζ) ε > C> S 1 (d, α ε/2, ρ ε) ν g (ζ,n) C N+1 Γ(1 + N/k 1 )., ĝ(ζ) C[[ζ]] 1/k1, g(ζ) S(d, α) exponential size k S 1 (d, α ε/2, ) { ζ ρ ε} C,h > ν g (ζ,n) C N+1 Γ(1 + N/k 1 )e h ζ k., δ > (z,ζ) S(d, α + π/k ε, δ) S 1 (d, α ε/2, ) (ζ/z) k < 2h ζ k., f(z) =L k,d (g)(z) (1.6) S(d, α + π/k ε, δ)., (1.7) ( ν f (z,n) =z N f(z) L ( N 1 )) k g n ζ n n= ν f (z,n) =z N L k,d (ζ N ν g (ζ,n))(z)., ν g (ζ,n) ν f (z,n) : S(d, α+ π/k ε, δ) C 1 >, N ν f (z,n) C N+1 Γ(1 + N/k 1 )Γ(1 + N/k)., L k,d (g)(z) = k2 Lk (ĝ)(z). 44

47 1.6. g(ζ) g (k) (ζ) =g(ζ 1/k ), L 1,kd L k,d L k,d (g)(z) =L 1,kd (g (k) )(z k )., ĝ(ζ) C[[ζ]] ĝ (k) (ζ) =ĝ(ζ 1/k ) C[[ζ 1/k ]], (1.7) L 1 C[[ζ 1/k ]] L k (g)(z) = L 1 (g (k) )(z k ). Borel Laplace : 1.1. f(z) S(d, α, ρ) (α > π/k) L k,d B k,d (f)(z) =f(z). Proof. δ > z = δe 2πid L k,d B k,d (f)(z) = 1 e id f( z)e ζk ( z k z k) z k z k d( z k )dζ k 2πi γ = 1 f( z)z k z e id k d( z k ) e ζk ( z k z k) dζ k 2πi = 1 2πi = k 2πi γ γ γ f( z)z k z k z k z k d( z k ) f( z) z k 1 z k z k d( z), δ > z γ., Cauchy. : g(ζ) S(d, α) (α > ), ζ = exponential size k B k,d L k,d (g)(ζ) =g(ζ). 1.5 Borel, Borel-Ritt : 45

48 1.12. d R, α (, π/k), ρ >., ˆf(z) C[[z]] 1/k f(z) = k ˆf(z) f(z) Ak (S(d, α, ρ)). Proof. g(ζ) D δ = {ζ C ζ δ} (δ > ). g(ζ) L δ k,d (g)(z) L δ k,d(g)(z) = δe id g(ζ)e (ζ/z)k z k dζ k., ˆf(z) C[[z]]1/k δ B k ( ˆf)(ζ) D δ. Sk,d δ ( ˆf)(z) :=L δ k,d B k ( ˆf)(z), Sk,d δ ( ˆf)(z) A k (S(d, α, ρ)) Sk,d δ ( ˆf)(z) = k ˆf(z)., Bk ( ˆf)(ζ) D δ, C> N, ζ D δ ˆB k ( ˆf)(ζ) N 1 f n ˆBk (z n )(ζ) C N+1 ζ N (1.8) n=., (1.5),(1.7), (1.8) : C 1 > z S(d, α, ρ) ( f n z n Sk,d(z δ n ) ) = e id δe id f n ζ n Γ(1 + n/k) e (ζ/z)k z k dζ k C n+1 Γ(1 + n/k) z n exp[ (δe id /z) k ], Lk,d( δ ˆBk ( ˆf)(ζ) N 1 ) e id f n ˆBk (z n )(ζ) C N+1 ζ N e (ζ/z)k z k dζ k n= C N+1 1 Γ(1 + N/k) z N. S(d, α, ρ) S δ k,d ( ˆf)(z) = k ˆf(z) α (, π/k) T k : A k (S(d, α, ρ)) C[[z]] 1/k., exp[ (e id /z) k ] A () k (S(d, α, ρ)) T k., α > π/k T k, Watson : ρ >, α > π/k A () k (S(d, α, ρ)) =. 46

49 Proof. f(z) A () k (S(d, α, ρ)) f(z) = k, 1.8 B k,d (f)(ζ) = B k ()(ζ) =., 1.1 f(z) =L k,d B k,d (f)(z) =L k,d ()(z) =. : ˆf(z) C[[z]]1/k,. (i) f(z) A k (S(d, α, ρ)) (ρ >, α > π/k) f(z) = k ˆf(z). (ii) ε > B k ( ˆf)(ζ) S(d, ε) exponential size k., ˆf(z) f(z) f(z) =L k,d B k ( ˆf)(z) (Borel ). ˆf(z) C[[z]]1/k 1.14 ˆf(z) d k-borel, k-summable, d k-summable C{z} k,d. ˆf(z) C{z} k,d, Borel S k,d ( ˆf)(z) S k,d ( ˆf)(z) :=L k,d B k ( ˆf)(z)., R mod 2π k-summable C{z} k., ˆf(z) C[[z]] 1/k k-summable ˆf(z), Sing( ˆf) ( R mod 2π). n d := C{z} k,d m = z C{z} k,d n := C{z} k m = z C{z} k : (C{z} k,d, n d ), (C{z} k, n) PID d, k> C{z} C{z} k,d k> C{z} C{z} k. 47

50 1.19. ˆfk (z) = Γ(1 + n/k)z n B k ( ˆf k )(ζ) =(1 ζ) 1., n= ˆf k (z) d/ 2πZ k-summable Sing( ˆf k )={ mod2π}., k l ˆf k (z) / C{z} l. C{z} k,d, C{z} k : 1.2. C{z} k,d, C{z} k C{z} a<b d (a, b), k>., ˆf(z) d (a,b)\{d }, k > B k ( ˆf)(ζ) S(d, ε) (ε > ) exponential size k ˆf(z) C{z} k,d Phragmén-Lindelöf : g(ζ) S = S(d, π/k) exponential size k ( k (,k) ). g(ζ) S S g(ζ) S. Proof. d =. sup g(ζ) M S, ε > l ( k, k) S e εζl g(ζ) M., S g(ζ) Me ε ζ l, ε >, S g(ζ) M k> C{z} = d R C{z} k,d. Proof. C{z} k,d C{z}. ˆf(z) C{z} k,d d R d R B k ( ˆf)(ζ) C exponential size k., ρ > S k,d ( ˆf)(z) {z C < z < ρ}., z = S k,d ( ˆf)(z) z = S k,d ( ˆf)(z) = k ˆf(z) C{z}. 48 C{z} k,d

51 1.24. k 1 >k 2 > C{z} = C{z} k1 C{z} k2 = C[[z]] 1/k1 C{z} k2. Proof. C{z} C{z} k1 C{z} k2 C[[z]] 1/k1 C{z} k2, ˆf(z) C[[z]] 1/k1 C{z} k2 B k ( ˆf)(ζ) C exponential size k 3 (k3 1 = k2 1 k1 1 )., 1.21, C[[z]] 1/k, B k C{ζ}., : 1.25 ( ). k, k 1 >., f(ζ),g(ζ) A k1 (S(d, α, ρ)) (α, ρ > ) f(ζ) g(ζ) f k g(ζ). [ d t f k g(ζ) = dt ] f((t t) 1/k )g( t 1/k )d t t=ζk 1.7. k 1 : f(ζ),g(ζ) A k1 (S(d, α, ρ)) f (k) (ζ) =f(ζ 1/k ), g (k) (ζ) =g(ζ 1/k ) : f k g(ζ) =f (k) 1 g (k) (ζ k ) f(ζ),g(ζ) A k1 (S(d, α, ρ)) (α, ρ > ) f(ζ) = k1 ˆf(ζ), g(ζ) = k1 ĝ(ζ)., h(ζ) :=f k g(ζ) h(ζ) A k1 (S(d, α, ρ)) h(ζ) = k ĥ(ζ), h n = n j= Γ(1 + (n j)/k)γ(1 + j/k) f n j g j (n ) Γ(1 + n/k)., [ d t ζ m k ζ n = (t t) m/k t ] n/k d t t=ζk dt 49

52 [ d ] = dt t1+(m+n)/k B(1 + m/k, 1+n/k) t=ζk = Γ(1 + m/k)γ(1 + n/k) ζ m+n Γ(1 + (m + n)/k), ν h (ζ,n), C{ζ} k Conv k, 1.26 : B k : C[[z]] 1/k Conv k Bk L k., Exp k,d C{ζ}, ε > S(d, ε) exponential size k k, 1.14 : B k : C{z} k,d Exp k,d., s B k ( z d dz zk+s )(ζ) =, ˆf(z) C[[z]] 1/k : (k + s)ζ k+s Γ(1 + (k + s)/k) = kζk Bk (z s ) B k ( z d dz ( z k ˆf(z) ) ) = kζ k Bk ( ˆf)(ζ). (1.9), 1.28 : f(z) C{z}, ĝ(z) n d, f(ĝ(z)) C{z} k,d. Proof. g B (ζ) := B k (ĝ)(ζ), g B (ζ) S(d, ε) (ε > ), S(d, ε) S 1 C, h > g B (ζ) C ζ e h ζ k. Cauchy, C, h, dg B /dζ S 1 dg B dζ (ζ) Ce h ζ k 5

53 ., n B k (ĝ n )(ζ) =gb n (ζ), g n B (ζ) Cn ζ n Γ(1 + n/k) eh ζ k (1.1)., n = n, g B () =, t g B k g n B (ζ) = t dg B dζ ((t t) 1/k ) 1 k (t t) 1/k 1 g n B ( t 1/k )d t t=ζ k Ce h( t t ) 1 k ( t t ) 1/k 1 Cn t n /k eh t d t Γ(1 + n /k) C n+1 ζ n+1 e B(1/k, 1+n /k) h ζ k Γ(1 + n /k) Cn +1 ζ n +1 Γ(1 + (n +1)/k) eh ζ k. t = ζ k, (1.1) B k (f(ĝ))(ζ) S 1 exponential order k. 1.7 Weierstrass C[[z]] 1/k, C{z} k,d, C{z} k Hensel. f(ζ,w), g(ζ,w) C{ζ,w}, f g(ζ,w) :, [ d t f g(ζ,w)= dt ] f((t t) 1/k,w)g( t 1/k,w)d t t=ζk. (1.11), Exp w k,d f(ζ,w) C{ζ,w}, ε >, S w S w = S(d, ε) {w C w < ε} f(ζ,w) Ce h ζ k. Exp w k,d. R C{ζ,w}, Exp w k,d., R f(ζ,w) R f(, ), R R., Weierstrass ( [GR] ), : 51

54 1.3. g(ζ,w) R j wg(, ) = ( j n 1), (1.12) n wg(, ) (1.13)., f(ζ,w) R q(ζ,w),r(ζ,w) R : f = q g + r, n wr(ζ,w)=. g(ζ,w) R (1.12), (1.13) w order n, w monic n g(ζ,w) Weierstrass., 1.3 : g(ζ,w) R w order n., n Weierstrass W (ζ,w), q(ζ,w) R, : g = q W., monic f(z,w) ( C[[z]] 1/k ) [w], f(,w) = ḡ(w) h(w)(ḡ, h C[w] ), 1.31 g(z,w),h(z,w) ( C[[z]]1/k ) [w] f(z,w) =g(z,w)h(z,w), g(,w)=ḡ(w), h(,w)= h(w). C{z} k,d, C{z} k, : C[[z]] 1/k, C{z} k,d, C{z} k Hensel. 1.9., Gevrey, Weierstrass. [Z] [Ro] C[[z]] Hensel C[[z]]. 1.8 Cauchy-Heine Cauchy-Heine,., Cauchy-Heine : 52

55 1.33 (Cauchy-Heine ). ψ(z) A () k (S(d, α, ρ)) (α, ρ > ), a S(d, α, ρ)., ψ(z) Cauchy-Heine CH a (ψ)(z). CH a (ψ)(z) : CH a (ψ)(z) = 1 2πi a ψ(w) w z dw ψ(z) A () k (S(d, α, ρ)) (α, ρ > ), a S(d, α, ρ), d = d + π, α = α +2π, ρ = a., CH a (ψ)(z) A k (S( d, α, ρ)) CH a (ψ)(z) = k ĈH a (ψ)(z), ĈH a (ψ)(z) := 1 2πi, z S(d, α, ρ) a z n w n 1 ψ(w)dw. n= CH a (ψ)(z) CH a (ψ)(ze 2πi )=ψ(z). Proof., CH a (ψ)(z) S( d, α, ρ), [,a] z S(d, α, ρ), CH a (ψ)(z) A k (S( d, α, ρ)), ψ A () k (S(d, α, ρ)) N 1 1 w z = z n w n 1 + zn w N w z n=,. Cauchy. : 1.35 ( ). S 1 = R /2π Z I j (1 j m), α j+1 < β j < α j+2 (1 j m) α j, β j R (1 j m), I j =(α j, β j )mod2π., j>m α j = α j m +2π., D r := {z C < z <r} (r >) S j (1 j m), α j, β j (1 j m) S j = S((α j + β j )/2, (β j α j )/2,r). 53

56 , 1.34 : S j (1 j m) D r, ψ j(z) A () k (S j 1 S j ) (S = S m )., a j S j 1 S j f j (z) = j CH al (ψ l )(z)+ l=1 m l=j+1 CH al (ψ l )(ze 2πi ) ( j m) (1.14), S j := S j Dρ (ρ =min a j ) f j (z) A k ( S j )( j m) j : f j (z) = k m l=1 ĈH al (ψ l )(z), f j (z) f j 1 (z) =ψ j (z), f m (ze 2πi )=f (z). k-summable : ˆf(z) C{z}k, Sing( ˆf) ={d j mod 2π 1 j m}., Sing( ˆf j )={d j mod 2π} ˆf j (z) C{z} k (1 j m) m ˆf(z) = ˆf j (z). j=1 Proof. d 1 < d 2 < < d m < d m+1 = d m +2π., d j (d j,d j+1 ) f j (z) = S k, dj ( ˆf)(z), f (z) = f m (z), ψ j (z) = f j (z) f j 1 (z) (1 j m), ψ j (z) A () k (S j 1 S j )., S j ε, ρ > 1.35 α j = d j π/2k +ε, β j = d j+1 +π/2k ε Dρ., a j = ρe 2πid j ( ρ (, ρ)), Sj α j = d j π π/2k + ε, β j = d j + π + π/2k ε D ρ 1.34 CH aj (ψ j )(z) A k ( S j )., ε >, CH aj (ψ j )(z) = k ĈH aj (ψ j )(z) C{z} k Sing(ĈH a j (ψ j )) = {d j mod 2π}., f j (z) (1.14) g j (z) := f j (z) f j (z), 1.34 S j 1 S j g j (z) g j 1 (z) =., g j (z) D ρ z =, h(z) =g j(z) C{z}., f j (z) +h(z) = f j (z) = k ˆf(z), ˆfj (z) =ĈH a j (ψ j )(z) (1 j m 1), ˆf m (z) =ĈH a m (ψ j )(z)+h(z). 54

57 1.9 k>1/2., S 1 Gevrey order k A k : I S 1. A k (I), ρ : I (, 1) S ρ := { ρe iθ < ρ < ρ(θ), θ I}. S 1 S 1, S 1 A k. [Ra2],[MR], A k., C[[z]] 1/k S 1, A k T k : A k C[[z]] 1/k., A () k := Ker T k, 1.12 : A () k A k C[[z]] 1/k. (1.15) Γ(S 1, A () k )=,Γ(S 1, A k )=C{z}, Γ(S 1, C[[z]] 1/k )=H 1 (S 1, C[[z]] 1/k )= C[[z]] 1/k, (1.15) : C{z} C[[z]] 1/k H 1 (S 1, A () k ) H 1 (S 1, A k ) C[[z]] 1/k. (1.16),, S : (1.16) H 1 (S 1, A () k ) H 1 (S 1, A k ). 1.38, : C{z} C[[z]] 1/k H 1 (S 1, A () k ). (1.17), H 1 (S 1, A k ) C[[z]] 1/k., I (d) k, : =[d π/2k, d + π/2k] mod2π ( A k )S 1 \I (d) k A k ( A k )I (d) k. (1.18), I (d) ι 1 k S 1 S 1 \ I (d) k ι 2 55

58 , I (d) k, ( A k )S 1 \I (d) k : A () k = ι 2! ι 1 2 A k, (1.18) Γ c (S 1 \ I (d) k ; A k) C{z} Γ(I (d) k ; A k) Hc 1 (S 1 \ I (d) k ; A k) H 1 (S 1, A k ) H 1 (I (d) k ; A k). (1.19) (1.19).,, Watson., Γ c (S 1 \ I (d) k T k : Γ(I (d) k ; A k) C{z} k,d Γ(I (d) k ; A () k )= ; A k)=,hc 1 (S 1 \ I (d) ; C[[z]] 1/k) C[[z]] 1/k, Hc 1 (S 1 \ I (d) k ; A () k ) Hc 1 (S 1 \ I (d) k ; A k) C[[z]] 1/k H 1 (S 1 ; A () k ) H 1 (S 1 ; A k )., : k Γ(I (d) k ; A k) C[[z]] 1/k C{z} C{z} k,d H 1 c (S 1 \ I (d) k ; A () k ). (1.2), S 1 I B k (I) I k-summable, S 1 k-summable B k, A k, B k :, Ŝ1 S 1, D k = {(θ 1, θ 2 ) S 1 Ŝ1 arg θ 1 arg θ 2 π/2k}., p 1 1, p 2 2 : p S 1 S 1 Ŝ1 p 2 56 Ŝ 1

59 , A k = p 2 (p 1 1 A k ) Dk, A k, B k : B k A k., G k := GL(n; A k ), Ĝ k := GL(n; C[[z]] 1/k ), T k : G k Ĝk, G () k := kert k., (1.15) : 1 G () k G k Ĝk 1. (1.21), Cartan 1.38 : (.2) H 1 (S 1, G () k ) H 1 (S 1, G k ) , [Si].,. [Ma], [H]. 1.1., Watson l>k>1/2 Proof., : Γ(I (d) k ; A () k /A () l )=. A () l A () k A () k /A () l., Watson Γ(I (d) k ; A () k )=Γ(I (d) k ; A () l )=, : Γ(I (d) k ; A () k /A () l ) H 1 (I (d) k ; A () l ) ι1 H 1 (I (d) k ; A () k ). 57

60 , ι : Γ(I (d) k ; A l) T l C[[z]] 1/l l H 1 (I (d) k ; A () l ) Γ(I (d) k ; A k) T k C[[z]] 1/k k ι 1 H 1 (I (d) k ; A () k ), ϕ Ker ι 1, l ( ˆf) =ϕ ˆf C[[z]] 1/l. ( (d), ˆf Im Tl Γ(I k ; A l) )., k ( ˆf) = ˆf ( (d) T k Γ(I k ; A k) ), l>k>1/2 T l : Γ(I (d) k ; A l) C{z} k,d C[[z]] 1/l. Proof. Watson.. ˆf C{z}k,d C[[z]] 1/l T k (f) = ˆf f Γ(I (d) k ; A k)., 1.8 k1 1 = k 1 l 1 S(d, π/2k 1 )\{} B l,d (f) = k1 Bl ( ˆf)., ˆf C[[z]]1/l B l ( ˆf) C{z}, T l B l,d (f) = Bl ( ˆf)., B l,d (f) S(d, π/2k 1 ) exponential order l, 1.9 f = L l,d B l,d (f) Γ(I (d) k ; A l), f = l ˆf.. = k m+1 >k m >k m 1 > >k 1 = k>1/2 k =(k m,k m 1,,k 1 ), d =(d m,d m 1,,d 1 ) R m I (dm) k m I (d m 1) k m 1 I (d 1) k 1., 1 j m A j := Γ(I (d j) k j ; A k /A () k j+1 ), B j := Γ(I (d j+1) k j+1 ; A k /A () k j+1 ) 58

61 ., : A 1 A B 1 B 2 A m B m 1 A 1 B1 A 2 B2 Bm 1 A m A k, d. f =(f1,,f m ) A k, d k-precise quasifunction, f j k j -precise quasifunction., A () l Ker T k (l k), T k : A j C[[z]] 1/k (1 j m),(f 1,,f m ) A k, d T k (f 1 )= = T k (f m )., (f 1,,f m ) A k, d T k (f 1 ) T k : A k, d C[[z]] 1/k, Im T k C{z} k, d, d k-summable., (f 1,,f m ), (g 1,,g m ) A k, d, (f 1 g 1,,f m g m ) A k, d, T k A () =, A m = Γ(I (dm) k m ; A k ), m = 1 T k : A 1 C{z} k1,d 1., : Ker T k =. Proof. f =(f1,,f m ) Ker T k., T k (f 1 )=, f 1 Γ(I (d 1) k 1 ; A () k /A () k 2 ), 1.4 f 1 =. B 1 f 1 = f 2, f 2 Γ(I (d 2) k 2 ; A () k 2 /A () k 3 ), 1.4 f 2 =., f j =(1 j m)., 1.42 T k : A k, d C{z} k, d, ˆf C{z} k, d T 1 k ( ˆf) =(f 1,,f m ), f m ˆf Borel, S k, d ( ˆf) f, g A k, d, f m = g m, f g Ker T k., 1.42 f m A k, d f. C{z} k, d, : 59

62 1.43. ˆf m C{z} k, d,g(z) C{z} g( ˆf) C{z} k, d., f =(f 1,,f m ) A k, d T k ( f )= ˆf, g( f )=(g(f 1 ),, g(f m )) A k, d T k (g( f )) = g( ˆf)., n k, d = m C{z} k, d, : ( C{z} k, d, n k, d ) PID. k-summable., ˆf (j) C{z} kj,d j (j =1,,m)., ˆf C{z} k, d., ˆf = ˆf (1) + + ˆf (m) (1.22) f m = S k1,d 1 ( ˆf (1) )+ + S km,d m ( ˆf (m) ) (1.23), f m A k, d, T k (f m )= ˆf. k-summable :, ˆf C[[z]]1/k : (i) ˆf C{z} k, d (ii) ˆf (j) C{z} kj,d j (j =1,,m) (1.22). Proof. (i) (ii). (f 1,,f m ) A k, d, T k (f m )= ˆf., B m 1 f m = f m 1, f m 1 I (d m 1) k m 1 {I j } p j=1 f m 1,j Γ(I j ; A k ) : q (1 q p) I (d m) k m I q, I j I (dm) k m =Ø (j q), f m 1,q = f m, f m 1,j f m 1,j+1 Γ(I j I j+1 ; A () k m )., Cauchy-Heine 1.36 g m 1,j Γ(I j ; A km ) I j I j+1 f m 1,j f m 1,j+1 = g m 1,j g m 1,j+1 6

63 ., h m 1,j = f m 1,j g m 1,j, I j I j+1 h m 1,j = h m 1,j+1 h m 1 Γ(I (d m 1) k m 1 ; A k )., g m 1,q =km ˆf (m) ˆf (m) C{z} km,d m., ĝ C[[z]] 1/km g j A j (1 j m 2) T k (g j )=ĝ, ĥ = ˆf (m) ˆf, k =(k 1,,k m 1 ), d =(d 1,,d m 1 ) ĥ C{z} k, d., ˆf (j) C{z} kj,d j (j =1,,m) ˆf (1.22) ˆf (j) C{z} kj,d j (j =1,,m), P (y 1,,y m ) C[y 1,,y m ] P ( ˆf (1),, ˆf (m) ) C{z} k, d. 1.45, ˆf C{z} k, d Borel S k, d ( ˆf) (1.23), J. Ecalle Acceleration, S k, d ( ˆf)., k >k> d ( k, k)-acceleration A k,k,d : A k,k,d (f)(ζ) =ζ k e id f( ζ)c k/k ( ( ζ/ζ) k ) d ζ k, C α (t) = 1 u 1 1/α u 1 tu 1/α e du 1 (α > 1). 2πi γ 1, C α (t) γ 1 (1.4) d =,k =1., A k,k,d, f(ζ) Exp k,d A k,k,d (f) A k,k,d (f)(ζ) =B k,d L k,d (f)(ζ) (1.24)., B k,d L k,d (f)(ζ) = = e id e (ζ/z) k z kdz k γ e id f( ζ)d ζ k γ e ( ζ/z) k z k f( ζ)d ζ k z k k e (ζ/z) k ( ζ/z) k dz k, u =(z/ζ) k (1.24)., f(ζ) =ζ s (s ) A k,k,d (f)(ζ) = 61 Γ(1 + s/k) Γ(1 + s/ k) ζs (1.25)

64 ., ˆf(ζ) = n f nζ n C[[ζ]]  k,k ( ˆf)(ζ)  k,k ( ˆf)(ζ) = n=., : α > 1 β β 1 =1 α 1 Γ(1 + n/k) Γ(1 + n/ k) f nζ n., ε > c 1,c 2 > S(, π/β ε). C α (t) c 1 e c 2 t β (1.26) Proof. C α (t) v = t α u C α (t) = t v 1 1/α t αv 1 v 1/α e 2πi γ 1 dv 1., f(v) =v 1/α e v 1/α C α (t) =tb 1, (f)(t α )., f(v) A () 1/α (S(, απ)), 1.8 B 1,(f)(ζ) A () α 1(S(, (α 1)π))., ζ = t α, 1.7 (1.26)., 1.46 : S = S(d, δ) (δ > ), k >k> κ κ 1 = k 1., f(ζ) A l (S) S exponential size κ, ε > ρ(ε) > S = S(d, δ +π/κ ε, ρ(ε)) A k,k,d (f)(ζ)., l 1 = l 1 + κ 1 A k,k,d (f)(ζ) A l( S). k 1 A k,k,d (f)(ζ) = l  k,k ( ˆf)(ζ). 62

65 1.15. A k,k,d L k,d., : ˆf C[[z]]1/k : (i) ˆf C{z} k, d (ii) k κ j (1 j m) κ 1 j = k 1 j k 1 j+1. g j (ζ) (1 j m) B k1 ( ˆf)(ζ) =g 1 (ζ), g j+1 (ζ) =A kj+1,k j,d j (g j )(ζ), ε > g j (ζ) S(d j, ε) exponential size κ j. Proof. (i) (ii) ˆf C{z} kj,d j., i < j g i (ζ) exponential size (k 1 i k 1 j ) 1, (k 1 i k 1 j ) 1 < κ i., g j (ζ) = B kj ( ˆf)(ζ), ˆf C{z}kj,d j g j (ζ) d j exponential size k j (< κ j )., g j+1 (ζ) =B kj+1,d j L kj,d j (g j )(ζ) g j+1 (ζ) d j+1 exponential size k j+1. i j +1 g i (ζ) d i exponential size k i. (ii) (i), f j (z) =L δ k j+1,d(g j+1 )(z) (1 j m 1), f m (z) =L km,d m (g m )(z)., L δ k j+1,d 1.12, d d j < ε/2+π/2κ j L δ k j+1,d δeid Gevrey order k j+1., f =(f1,,f m ) A k, d, T k ( f )= ˆf., f j A j T k (f j )= ˆf, B j 1 f j = f j 1 :, g j (ζ) A δ k j+1,k j,d j (g j )(ζ) A δ k j+1,k j,d j (g j )(ζ) =B kj+1,d j L δ k j,d j (g j )(ζ) 63

66 ., gj+1(ζ) δ =A δ k j+1,k j,d j (g j )(ζ), 1.46 g j+1 (ζ) gj+1(ζ) δ Gevrey order k j k j+1 /(k j+1 k j ), L δ k j+1,d j+1 (g j+1 gj+1)(z) δ Gevrey order k j+1., 1.1 f j 1 (z) =L kj+1,d j (gj+1)(z) δ, f j 1 (z) L δ k j+1,d j+1 (gj+1)(z) δ Gevrey order k j+1. B j 1 f j = f j 1., ˆf C{z} k, d : ˆf C{z} k, d, Borel S k, d ( ˆf) : S k, d ( ˆf) =L km,d m A km,k m 1,d m 1 A k2,k 1,d 1 B k1 ( ˆf). 2 n A(z) M(n; C{z}), k, : z k+1 d ϕ = A(z)ϕ. (2.1) dz (2.1) z =,, [BJL], [Ra2], [Ba2],, Borel. 2.1 Splitting Lemma, (2.1) A() λ 1,, λ l. P GL(n; C) ϕ = P ϕ A() Jordan. λ j Jordan J(λ j ) M(n j ; C)., l =2 : ( ) J(λ1 ) A() = (λ J(λ 2 ) 1 λ 2 )., ϕ = T (z)ψ ( T (z) GL(n; C{z} k ) ), (2.1) : z k+1 d ψ = B(z)ψ, (2.2) dz 64

67 ( ) B1 (z) B(z) =. B 2 (z), B j (z) M(n j ; C{z} k )(j =1, 2) B j () = J(λ j )., (2.1) ϕ = T (z)ψ ψ z k+1 d ( dz ψ = T 1 z k+1 d )ψ dz T + AT., (2.2), T (z), B(z) z k+1 d T = AT TB (2.3) dz., T (z), B(z) M(n; C[[z]] 1/k ) (2.3)., T (z) : T (z) =I n + ( T12 (z) T 21 (z), I n n, T ij M(n i,n j ; z C[[z]]). A(), A(z) ( ) A11 (z) A A(z) = 12 (z) (2.4) A 21 (z) A 22 (z)., (2.3) (1, 1). (2, 1) ). =A 11 + A 12 T 21 B 1 (2.5) z k+1 d dz T 21 = A 21 + A 22 T 21 T 21 B 1 (2.6)., (2.5) (2.6) B 1, T 21 : z k+1 d dz T 21 = A 21 + A 22 T 21 T 21 A 11 T 21 A 12 T 21. (2.7) B 2, T 12 (2, 2), (1, 2). 65

68 T 21. T 21, A ij z p T (p) 21, A (p) ij : T 21 = A ij = p=1 p= T (p) 21 z p, A (p) ij zp., (2.7) z p : (p k)t (p k) 21 = A (p) 21 + p q=1 A (p q) 22 T (q) 21 p 1 A () 22 T (p) 21 + T (p) 21 A () 11 =A (p) 21 + q=1 q 1 +q 2 +q 3 =p q 1,,q 3 1 p q=1 T (q) 21 A (p q) 11 p 1 A (p q) 22 T (q) 21 q=1 q 1 +q 2 +q 3 =p q 1,,q 3 1 T (q) 21 A (p q) 11 T (q 1) 21 A (q 2) 12 T (q 3) 21. T (q 1) 21 A (q 2) 12 T (q 3) 21 (p k)t (p k) 21. (2.8), (2.8) T (q) 21 (1 q p 1), T (p) 21., λ 1 λ 2, A () 22 + A () 11 M(n 2,n 1 ; C) L., (2.8) T (p) 21 (p 1). T 21 M(n 2,n 1 ; z C[[z]] 1/k ). P =(p ij ) M(p, q; C), P P = i,j p ij., A(z) M(n; C{z}), C > p A (p) ij Cp+1., : C>, p 1 T (p) 21 C p Γ(p/k). (2.9) 66

69 p =1, 1 p p 1 (2.9) p = p (2.9). C>C., (2.8) 2 p 1 A (p q) 22 T (q) 21 q=1 p 1 q=1 C C p ( C C C C C p Γ ) p q Γ(l/k) C ( p 1 )., (2.8) 5 ( ) (p k)t (p k) 21 C p k p k (p k)γ k ( = kc p k p ) Γ k., L 1, C> p 1 (2.9), T 21 M(n 2,n 1 ; z C[[z]] 1/k ). T 21 M(n 2,n 1 ; z C{z} k )., X := B k (T 21 ), Ã ij,b := B k (A ij A () ij ), (1.9),(2.7) T 21,B : A () 22 X+XA () 11 +kζ k X = kζ k X+Ã21,B+Ã22,B X X Ã11,B X Ã12,B X. (2.1), (2.1) A () 22 + A () 11 + kζ k L ζ, S {ζ C kζ k = λ 2 λ 1 } =Ø S = S(d, α) L ζ., Exp h k( S) S, S f(ζ) f := sup e h ζ k f(ζ) < ζ S, Exp h k( S) Banach. P (ζ) = (p ij (ζ)) = M(p, q; Exp h k( S)) P P = ( p ij ). : C, h > S k Ãij,B C e h ζ k (2.11). h>h, (2.1) 3 Ã22,B() = Ã22,B X t dã22,b dζ ((t t) 1/k ) 1 k (t t) 1/k 1 X( t 1/k )d t t=ζ k 67

70 t C k 1 e h( t t ) ( t t ) 1/k 1 X e h t d t t=ζ k C k 1 (h h ) 1/k e h t X., (2.11) dã22,b/dζ (2.11). (2.1) 5 X Ã12,B X C k 2 (h h ) 1/k e h t X 2. (2.1)., F (X) =L 1 ζ (kζ k X + Ã21,B + Ã22,B X X Ã11,B X Ã12,B X), L 1 ζ M(n 2,n 1 ; Exp h k( S)), h, M>, F B M := {X M(n 2,n 1 ; Exp h k( S)) X M}., Banach (2.1) X M(n 2,n 1 ; Exp h k( S)), B k (T 21 ). T 21, B 1 d =(arg(λ 2 λ 1 )+2π Z)/k k-summable. T 12, B 2 d =(arg(λ 1 λ 2 )+2π Z)/k k-summable., : A () λ 1,, λ l, λ i λ j (i j), J(λ 1 ) A () J(λ 2 ) =... J(λ l )., B j (z) M(n j ; C{z} k ), T ij (z) M(n i,n j ; z C{z} k ) = J(λ j ), B () j B 1 (z) B(z) = B 2 (z)... B l (z) 68

71 T 12 T 1l T 21 T 23. T (z) =I n T(l 1)l T l1 T l(l 1), ϕ = T (z)ψ (2.1) (2.2)., X ij, B j { 1 k arg(λ p λ j )+ 2π Z } k p j k-summable., (2.3) A ij, B j, T ij B j = A jj + p j A jp T pj, z k+1 d dz T ij = p A ip T pj T ij B j, T ij l =2 L J(λ i ) + J(λ j ) : X =(X 1,,X n ), F (z,x) =(F 1,,F n ) C n {z,x}. F (z,x), F (, ) = J = ( Xi F j (, ) ) J., z k+1 d dz X = F (z,x) X(z) C n [[z]] X(z) C n {z} k., J λ j (1 j n), X j n j=1 k-summable. { 1 k arg(λ j)+ 2π Z } k 69

72 2.2, A(z) M(n; C{z} k ) A () = J(λ) (2.1)., ϕ = e λk 1 z k ϕ λ =., ϕ = T (z)ψ ( T (z) GL(n; C{z} k ) ) (2.1) (2.2), M,N > B(z) = N+M p= B (p) z p, B (p) = A (p) ( p N 1). T (z) : T (z) =I n + T (p) z p., T (z) T (z) GL(n; C[[z]] 1/k ). X := B k (T I n ), Ã B := B k (A A () ), BB := B k (B B () ) (2.3) : A () X + XA () + kζ k X = kζ k X + ÃB B B + ÃB X X B B. (2.12), M(n; C) A () + A () L, A () + A () + kζ k L ζ, det(λ L) =λ µ f(λ) (f() ), ζ kµ L ζ ζ = D ρ := {ζ C ζ ρ}., Ã B D ρ., B 1 D ρ, D ρ n X(ζ) = ( x ij (ζ) ) X 1 := sup D ρ, B 2 N+M p=n p=n ζ N x ij (ζ) < i,j M(n; C)ζ p 7

73 , Y (ζ) = ( y ij (ζ) ) B 2, Y 2 := sup ζ N y ij (ζ) D ρ, B 1, B 2 Banach., tr A(z) = i,j N 1 p= A (p) z p, tr à B := B k ( tr A A () ), BB tr à B B 2., (X, Y ) B 1 B 2, ( F 1 (X, Y )=L 1 ζ G(X, Y ) F2 (X, Y ) ),., F 2 (X, Y )= 1 G(X, Y )( ζ)k N,M (ζ/ ζ) d ζ 2πi ζ =ρ ζ G(X, Y )(ζ) =kζ k X + ÃB tr à B + ÃB X X tr à B X Y, K N,M (t) =t N 1 tm+1 1 t., F 2 (X, Y ) B 2., G(X, Y ) G(X, Y ) ζ N, F 2 (X, Y ), G(X, Y ) F 2 (X, Y ) ζ N+M+1., M M kµ 1 F 1 (X, Y ) B 1., B = B 1 B 1, F =(F 1,F 2 ), N,C > F B C := {(X, Y ) B X 1 + Y 2 C}., C > sup D ρ dãb dζ., : ÃB X t <C dãb dζ ((t t) 1/k ) 1 k (t t) 1/k 1 X( t 1/k )d t t=ζ k 71

74 t C k 1 X 1 ( t t ) 1/k 1 t N/k d t t=ζ k C k 1 X 1 ζ N+1 B(1/k, 1+N/k). kζ k X, X tr à B., X Y X Y k 1 X 1 Y 2 ζ 2N B(N/k, 1+N/k)., B(1/k, 1+N/k), B(N/k, 1+N/k) (N ), ζ = ρ, N,C > F B C., Banach, T (z), B(z)., T (z) A(z) k-summable. A(z) d k-summable., < ρ < ρ ζ = ρe id, S := ζ + S(d, ε), ε >, à B S, C, h > : ÃB C e h ζ k. (2.13) B B S (2.13)., Exp h k( S), X M(n; Exp h k( S)) F (X) =L 1 ζ (ÃB B ( d t ) ) B + R + K(t t)x( t 1/k )d t dt t=ζ k, (2.14) t., K(t) = ti n +ÃB(t 1/k ) B B (t 1/k ), ( d t R(ζ) = K(t t) dt T ) B ( t )d t) 1/k t=ζ k, t =ζ k t =ζ k, TB = B k (T I n ), BB := B k (B B () ), T (z), B(z)., L 1 ζ M(n; Exp h k( S)), h, C >, F {X M(n; Exp h k( S)) X C}., Banach F (X) =X X M(n; Exp h k( S)), X T B S., T (z) A(z) k-summable. 72

75 , 2.2, (2.1) summable,. A(z) M(n; C[z]) J(; n 1 ) A () J(; n 2 ) =... J(; n l ) (2.1) i) k, ii) n., J(; n j ) M(n j ; C) n j Jordan., (2.1) k =, n =1, n =1. l =1. n 2., T (z) =I n + T (z), T (z) = N T (p) z p p=1 ψ = T (z)ϕ, N> (2.1) (2.2) B(z) =J(; n)+ N B (p) z p + B(z), (2.15) p=1. B (p) =...., (2.16) b (p) n1 b (p) nn B(z) M(n; z N+1 C[z]) ). (2.3) z p, T, B : J(; n)t (p) T (p) J(; n) =B (p) A (p) + R (p). (2.17) 73

76 , R (p) A (q), T (q), B (q) (1 q p 1)., T (p) =(t (p) ij ) t (p) 21 t (p) 22 t (p) t (p) 2n 11 t (p) 1(n 1) J(; n)t (p) T (p). J(; n) =..... t (p) n1 t (p) n2 t (p) t (p) 21 t (p) 2(n 1). nn..... t (p) n1 t (p) n(n 1)., T (p),, (2.16) B (p),(2.17). t (p) 1j =(1 j n) T (z)., B(z) (2.15). tr B(z) =B(z) B(z), b j (z) = N p=1 b (p) nj zp (1 j n)., P (z,ζ) =det ( ζ tr B(z) ) n 1 = ζ n + ( 1) n j b j+1 (z)ζ j j= det ( ζ B(z) ) = P (z,ζ) modz N+1 C[z,ζ]., P (z,ζ) Newton polygon, F r Q, ζ = z r ζ, P (z, ζ) n 1 =z nr P (z,z r ζ) = ζn + ( z r ) n j b j+1 (z) ζ j j=., F ζ n, ( 1) n j b j +1(z)ζ j b j +1(z) z = v,, r = v n j (2.18). r = p /q (p,q ). r P (z, ζ) Newton polygon F, 74

77 ., S r (z), Shearing ψ = S r (z) ψ : 1 z r S r (z) =.... z r(n 1), Shearing (2.2) : z k r+1 d dz ψ = C(z) ψ, (2.19). C(z) =J(; n)+.... r zk r... + C(z). b1 (z) bn (z) r(n 1), b j (z) =z r(n j+1) b j (z)(1 j n), C(z) z N rn+1 M(n; C[z 1/q ]) ). k r (2.19), k r>., r C(z) M(n; C[z 1/q ]) ), b j +1(), bj+1 () = (1 j j 1), N rn+1 > N, C(). C(), 2.1, 2.2 n., C() P (, ζ) = n, b j () (1 j n), P (z,ζ) Newton polygon F, (j 1,r(n j +1)) Z 2., r Z k Shearing n., q,(2.18) q n. l = 2. A(z) A() (2.4)., N A 21 (z) z N M(n 2,n 1 ; C[z]) )., (2.1) A = A 11 l =1 T (z), S r (z) T 1 (z), S r1 (z), A = A 22 T 2 (z), S r2 (z), ( ) T1 (z)s T (z) = r1 (z) z r 1(n 1 1) T 2 (z)s r2 (z) 75

78 , N r 1 n 1 r 2 n 2 >, l =1, n k., N. n 1 n 2, A (N) 21 (N 1)., ( ) (T21 T (z) =I n + z N M(n T 21 2,n 1 ; C) ) ϕ = T (z)ψ (2.1) (2.2) ( ) B11 (z) B B(z) = 12 (z), B 21 (z) B 22 (z) B () jj = J(; n j ) (j =1, 2), c 1 B 21 (z) = z N mod z N+1 M(n 2,n 1 ; C[z]) (2.2) c n2., (2.3) z N, T (N) 21, B (N) 21 M(n 2,n 1 ; C), (2.2) : J(; n 2 )T N 21 T (N) 21 J(; n 1 )=B (N) 21 A (N) 21. (2.21) (2.17), T (N) 21, B (N) 21. B (N) 21 = N N +1. B (N) 21, S(z) Shearing ψ = S(z) ψ : ( In1 S(z) = z N I n2, (2.2) ). z k+1 d dz ψ = C(z) ψ, ( ) J(; C () n1 ) = B (N) 21 J(; n 2 )., C (), ( C ()) j (j 1), c m,c j =(m +1 j n 2 ), C () Jordan (J(; ) n1 + m) J(; n 2 m) 76

79 ., n 1 n 2, m 1, l =1, N,, k n. l 3 k n.,,, M M(n; C), T (z) GL(n; C{z}[z 1 ]) (2.1) k = ϕ = T (z)ψ z d dz ψ = M ψ. M Jordan, [, 1). 2.4,., Λ := {λ i (ζ)} l i=1 C{ζ}[ζ 1 ]/ C{ζ}, G : L k = C{ζ}[ζ 1 ]/ζ k+1 C{ζ},., ρ k : L k L k+1 { λi (ζ) modζ k C{ζ} } l k=1 i=1 ( ) L k node, ρ k edge, {λ i (ζ)} l i=1 leaf G., L k node level k node., G edge, 2 node G., λ i (ζ) = λ (k) i ζ k mod C{ζ} k, G λ i (ζ) level k node (k, λ (k) i ) G Λ Λ := {λ i (ζ)} 2 i=1 k=1 λ 1 (ζ) =α (3) ζ 3 + α (2) ζ 2 + α (1) ζ 1 λ 2 (ζ) =β (3) ζ 3 + β (2) ζ 2 + β (1) ζ 1 77

80 , G Λ (, ) (3, α (3) ) (3, β (3) ) 2.2. Λ := {λ i (ζ)} 4 i=1, G Λ λ 1 (ζ) =α (4) ζ 4 + α (3) ζ 3 + α (2) ζ 2 + α (1) ζ 1 λ 2 (ζ) =β (4) ζ 4 + β (2) ζ 2 + β (1) 1 ζ 1 λ 3 (ζ) =β (4) ζ 4 + β (1) 2 ζ 1 λ 4 (ζ) = γ (3) ζ 3 + γ (2) ζ 2 + γ (1) ζ 1 (, ) (4, α (4) ) (4, β (4) ) (4, ) (2, β (2) ) (2, ), : q, ζ = z 1/q, Λ = {λ j (ζ)} l j=1 ζ 1 C[ζ 1 ],T(ζ) GL(n;(C[[ζ]] 1/kq )[ζ 1 ]), M j M(n j ; C) (j =1,, l), (2.1) ζ = z 1/q ϕ = T (ζ)ψ : B 1 (ζ) ζ d dζ ψ = B 2 (ζ)... ψ, B l (ζ) B j (ζ) =λ j (ζ)i nj + M j., M j Jordan, [, 1). -Levelt Turrittin., T (ζ) : T (ζ) =T (m) (ζ)t (m 1) (ζ) T (1) (ζ)t () (ζ), 78

81 T (i) (ζ) = T (i) 1 (ζ) T (i) 2 (ζ)..., T (i) (ζ) l (i+1) T (i) j (ζ) GL(n (i+1) j ; C{ζ} ki ) (1 i m, 1 j l (i+1) ), T () j (ζ) GL(n (1) j ; C{ζ}[ζ 1 ]) (1 j l (1) )., {k i } m+1 i=1 (k i+1 >k i, k m+1 = ) G Λ node level, {n (i) j }l(i) j=1 G Λ level k i n,,, n (i) j., G Λ Λ mod ζ ki+1 C{ζ} = { λ (i) j (ζ)}l(i) j=1 λ j1 (ζ) modζ k i+1 C{ζ} = λ (i+1) j (i) λ j (ζ) j 1 n j1 (ζ), edge edge level k i node T (i) j., node {λ (k i) Sing ki := j 1 j 2 j, j 1 } l(i) j j 1 =1 { 1 arg(λ (k i) j, j k 1 λ (k i) j, j 2 )+ 2π Z } i k i (ζ) =I n (i+1) (i), T j (ζ) k i -summable., A() Jordan,, {n (i) j, j 1 } l(i) j j 1 =1 G Λ level k i n (i+1) j, T (i) j (ζ) T (i) j, j 1 j 2 (ζ) M(n (i) j, j 1,n (i) j, j 2 ; C{ζ} ki ) T (i) (i) j, 11 (ζ) T (ζ) j, 1l (i) T (i) j j (ζ) =....., T (i) (i) T (ζ) j, l (i) 1(ζ) j, l (i) l(i), T (i) j, j 1 j 2 (ζ) p j 2 k i -summable. j { 1 arg(λ (k i) j, p k λ(k i) j, j 2 )+ 2π Z } i k i 79 j j j

82 q n! Levelt-Turrittin λ j (ζ), M j Jordan., T (ζ), T (ζ)., B(ζ), B(ζ), S(ζ) :=T 1 (ζ) T (ζ) : ζ d dζ S = BS S B. (2.22), S(ζ) GL(n;(C[[ζ]] 1/kq )[ζ 1 ]) (2.22) -Levelt-Turrittin., S., d =(d m,,d 1 ), d i / Sing ki, T (i) (ζ) Borel S ki,d i (T (i) ), ζ d dζ ψ = M jψ ζ M j (2.1) d S km,d m (T (m) ) S k1,d 1 (T (1) )T () ζ M 1... ζ Ml ([Br1]).,. [Br2] Acceleration, [RS] Cohomological [Ma]., [Mo], [Sa1], [Sa2]. References [Ba1] W. Balser: From divergent power series to analytic functions, Lecture Notes in Mathematics, Vol. 1582, Springer-Verlag,

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