k n ( k) n k F n N n C F n F n C F F q n N C F n k 0 C [n, k] [n, k] q

C [n, k] k n C C (n k) n C u C u T = T. [n, k] C (n k) n T = k (n k). F n N u = (u 1,..., u n ) v = (v 1,..., v n ) F n d(u, v) u v d(u, v) = {i {1,..., n} u i v i }. C [n, k] q k 1 d(c) = {d(u, v) u, v C, u v} C k = 0 d(c) = n + 1 d = d(c) [n, k, d] [n, k, d] q C. F n N u = (u 1,..., u n ) F n w(u) u 0 = (0,..., 0) F n. w(u) = {i {1,..., n} u i 0} = d(u, 0), d F n F n N. t t u F n F n N u r 0 S(u, r) = {v F n d(u, v) r}. C r N C r u C S(u, r) C = {u} C r u, v C S(u, r) S(v, r) =. C d r N C r r < d C r 2r < d

C r r r C r 2r < d 2r d u v 2r w F n d 2 u d 2 v d(u, w) = d 2 d(v, w) = d 2 u u v v d 2r d 2 r r u v w C r C r w F n u, v C w S(u, r) S(v, r). d d(u, v) d(u, w) + d(v, w) = r + r = 2r 2r < d [n, k, d] q d n k + 1 C F n [n, k, d] q C C C d 1 d 1 C d C C C F n (d 1) F = q. q k = C = C F n (d 1) = q n (d 1). [n, k, d] d = n k + 1 C [n, k, d] d 1 r N 0 r r d = 1 r 1 n

r = n n+1 u = (u 1,..., u n ) C J = {i {1,..., n} u i 0} h T 1,..., h T n u T = T i J ht i u i = 0 d d 1 r v r +1 v T = T v C d r + 1 C (c 1,..., c n ) C (c n, c 1,..., c n 1 ) C F n N { n 1 } F [x] n = c i x i c i F + a F n 1 n 1 n 1 c i x i + d i x i = (c i + d i )x i, n 1 n 1 c i x i = c i x i, n 1 n 1 a c i x i = ac i x i. b : F n F [x] n b((c 0,..., c n 1 )) = n 1 c ix i b b F [x] n x n 1 F [x] {}}{ p q = (p q) x n 1, F [x] n Π n : F [x] F [x] n Π n (p) = p x n 1 F [x] n F [x]/ Π n = F [x]/(x n 1). I F [x] n [0] (x n 1) Π 1 n (I) F [x] x n 1 F [x]

g x n 1 Π 1 n (I) I = (gf [x]) F [x] n F n N F [x] n C(g) = {gh h F [x], h < n g} g F [x] x n 1 g C(g) F [x] n g x n 1 C(g) F [x] n C(g) 0 C(g) r F [x] n gh C(g) rgh (x n 1) = g(rh ( xn 1)) C(g) g C F [x] n C g F [x] x n 1 C(g) = C. C F [x] n F u xu, x 2 u,..., x n 1 u C u C f = n 1 f ix i F [x] n n 1 f u = f i C {}}{ x i u }{{} C C, C F [x] n C = C(g) g x n 1 C(g) F [x] n F n h C(g) xh C(g)

R (R, +,,,, ) (R, +,, ) x + y = y + x x, y R (x + y) + z = x + (y + z) x, y, z R x + = + x = x x R x + (x) = (x) + x = x R x (y z) = (x y) z x, y, z R x (y + z) = x y + x z (y + z) x = y x + z x x, y, z R, x, y R x y = y x, x R x = x = x. F = (F, +,,,, ) x F x 1 F x x 1 = F < p m N p m F p m F R

f F [f] R [f] f f f [f] R F f F [f] R f F f = a 0 + r 0 = a 0 + a 1 + r 1 = a 0 + a 1 + a 2 + r 2 = a 0 + a 1 +, a n 1 + a n + r n n = 0, 1,... a n r n a 0 = [f], [ a n = r n 1 ] n 1 r n 1, r 0 = f a 0, r n = a n r n 1 n 1 r n 1. N N 0 r N = a n r n n N + 1 f F R n n 0 n 0 n N

n f = A n(a n + r n ) + B n C n (a n + r n ) + D n, A n B n C n D n a 0,..., a n 1 n n = 0 f = a 0 + r 0 = (a 0 + r 0 ) + (a 0 + r 0 ) + n 1 n 1 A n 1 B n 1 C n 1 D n 1 f = A n 1(a n 1 + r n 1 ) + B n 1 C n 1 (a n 1 + r n 1 ) + D n 1. an+rn a n+r n r n 1 = a n+r n f = A n 1(a n 1 + a n +r n ) + B n 1 C n 1 (a n 1 + an + r n a n +r n ) + D n 1 a n + r n = A n 1a n 1 (a n + r n ) + A n 1 + B n 1 (a n + r n ) C n 1 a n 1 (a n + r n ) + C n 1 + D n 1 (a n + r n ) A n B {}}{{}} n { = ( A n 1 a n 1 + B n 1 )(a n + r n ) + A n 1. (C n 1 a n 1 + D }{{ n 1 )(a } n + r n ) + C }{{ n 1 } C n D n A n = A n 1 a n 1 + B n 1, B n = A n 1, C n = C n 1 a n 1 + D n 1, D n = C n 1. A n B n C n D n r n = n f f n f n = P n Q n, P n = A n a n + B n, Q n = C n a n + D n.

A n B n C n D n P n Q n A n+1 = P n, B n+1 = P n 1, C n+1 = Q n, D n+1 = Q n 1, P n+1 = a n+1 P n + P n 1, Q n+1 = a n+1 Q n + Q n 1 0 n N P 0 = a 0, Q 0 =, P 1 =, Q 1 =. (2.2) (2.5) a n (2.11) (2.12) P n Q n f n f R P n Q n R P n Q n 0 n N P n Q n 1 Q n P n 1 = ( ) n+1. P n Q n 0 n ( N) ( ) ( ) ( ) Pn P n 1 Pn 1 P = n 2 an Q n Q n 1 Q n 1 Q n 2 ( ) ( ) ( ) Pn 2 P = n 3 an 1 an Q n 2 Q n 3 ( ) ( ) ( ) a0 a1 an =. P n Q n R f F n n Q n = f f n = f P n Q n.

f F P, Q R f = P /Q P 1 Q 1 1 n 1 P 0 [f] Q 0 0 f [f] n 0 n 0 n n [ + 1 ] a n n 2 n 1 P n a n P n 2 + P n 1 Q n a n Q n 2 + Q n 1 n a n n 2 + n 1 P = P n Q = Q n P Q n P n Q n n = fq n P n. {Q n } {P n } n+1 = a n+1 n + n 1, 1 =, 0 = f a 0 = r 0. (a n + r n ) A n, B n, C n D n a n + r n = f D n B n A n f C n = f Q n 2 P n 2 f Q n 1 P n 1 = n 2 n 1, a n = [ ] n 2, n 1. n 1 {a n } n (2.19) n = n 1 (2.5) = n n 1 = n 1 r n = r 0 ( r i ). a n+1 + r n+1 r n r N = 0 N = 0 a n a n = [ n 2 n 1 ] i=1

f = 5457 1853 = 2,94495412... n P n Q n n a n 1 1 0 1 1751 0 2 1 1853 [ ] 1 3 1 102 1 = [ 1853 1853 0,94495412... 1751] = 1 [ ] 17 2 53 18 0,94495412... = [ ] 1751 1853 0,0550458... 102 = 17 3 321 109 0 [ 0,0550458... 0,00917431... ] = [ 102 17 ] = 6 f f = 321 109 = 2 + 1 1 + 1. 17 + 1 6 F { } F ((x)) = f d i x d i f d 0, f d i F i 0 d Z = 0x = 1x 0 f d i x d i + g e i x e i = (f {d,e} i + g {d,e} i )x {d,e} i, x 1 F ((x 1 ))

f d i x d i h m i = g e i x e i = j+k=m i h m i x m i, i > d j > e f i = 0 g j = 0 f j g k, m = d + e, m = 0 h 0 = 1 h i = 0 i = 1, 2,... F ((x)) { d } F [x] = f d i x d i f d 0 F F [x] F ((x)) F ((x)) [ ] d f d i x d i f d ix d i, d 0, = 0, d < 0. F F 16 = Z2 /(α 4 + α + 1) R f(x) = x5 + α 9 x 4 + α 6 x 3 + α 9 x 2 x 3 + α 6 x 2 + α 3 x + α 13 = x 2 + α 5 x + α 9 + α 4 x 1 + 0x 2 + 0x 3 + α 2 x 4 +.... f(x) R f(x) F f(x) [f(x)] = x 2 + α 5 x + α 9. F F : F R + A B = A B A = 0 A = A + B ( A, B ).

F ((x)) f(x) F ((x)) d f(x) = 2 d = 0 F = 1 A 1 = A 1 A = A B > A A + B = B 0 x = N1) = = x x x = 1 1 V = 1) = A A 1 N1) = A A 1 A 1 = A 1 A 2 = A 2 = ( A) 2 = A 2 A = A B > A B = (A + B) A ( A + B, A ) = A + B R R F f F f [f] < 1 f F. f 1 f = [f] f = 2 m m f [f] 1 2 f = [f] ( f, [f] ) V = 4) f [f] N5) < 1 N3) ( f, [f] ) f = [f] F f 2 m < f < 2 m+1 m Z d F 1 < fd m V = 5) [fd m ] < 2 f P n = n Q n. Q n

F 1 2 n n = r i 2 (n+1). Q n [ ] a n = 1 = 2 n 1 r n 1 r n 1 Q n 2 < Q n 1 Q n 2 < a n Q n 1 Q n = a n Q n 1 2 Q n 1. Q 1 = 0 Q 0 = 1 { Q n } n 1 n Q n 2 n. 0 f P n = n Q n 1. 22n+1 Q n {P n /Q n } f N N N = 0. f = P N Q N {P n /Q n } f F P Q R n Q n Q f P n f P Q Q n f P n < 1 Q n Q. Q n r p/q q > 0 r

Q 1 = 0 { Q n } n Q n Q < Q n+1. f P /Q < f P n /Q n P Q P n Q n (f = P ) ( f P ) n Q Q n (V 4) = f P ( n (V 4) Q n = f P ) ( n f P ) n+1 Q n Q n+1 = P n Q n+1 P n+1 Q n Q n Q n+1 (2.13) = Q n Q n+1 < Q n Q. Q n Q P Q n P n Q <. P Q P n Q n R R P Q n P n Q P Q n P n Q = 0 P Q n P n Q =. f f F P n /Q n n A B R A AP n = BQ n. A Q n. P n Q n 1 Q n P n 1 = ±. A AP n Q n BQ n 1 AP n 1 = A. BQ n 1 AP n 1 R f(x) F ((x)) L L 1 f(x) = f d i x d i + X d L, d f(x) f d i F i = 0, 1,..., L 1 X d L F ((x)) d L f(x) F ((x)) F [x]

f f = x 2 + α 5 x + α 9 + α 4 x 1 + 0x 2 + 0x 3 + α 2 x 4 + α 8 x 5 + α 12 x 6 + X 7. f n P n Q n n a n 1 1. 0 1. 0 1α 5 α 9. 1..α 4 00α 2 α 8 α 12 X 7 1α 5 α 9. 1 α 11 αα 5 1. α 11 0..00α 13 α 4 α 8 X 6 α 11 0. 2 α 2 α 11 α 8 α 11 00. α 2 α 8 α 5 1..000X 4 α 6 α 12 α 9. f 2 = P 2 = α2 x 5 + α 11 x 4 + α 8 x 3 + α 11 x 2 Q 2 α 2 x 3 + α 8 x 2 + α 5 x + 1 = x5 + α 9 x 4 + α 6 x 3 + α 9 x 2 x 3 + α 6 x 2 + α 3 x + α 13 = f. f P n /Q n Pm/Q m K 1 f f F f P n Q n = f P m = Q m P,Q: Q K P,Q : Q K f P Q, f P. P n = P m, f f < Q n Q m Q 1 Q n K. P n /Q n = Pm/Q m. f f = f P n f + P m Q n Q m ( f P n Q n, f P ) m Q m ( ) 1 < Q n K, 1. Q m K

Q mp n = PmQ n Q m Q n f f 1 < Q n K, f f 1 < Q n K. Q mp n PmQ n = P n Q n Q P m m Q n Q m = f f f + P n + f P m Q n Q m ( f f, f P n Q n, f P ) m Q m ( ) 1 < Q n K, 1. Q m K Q n Q m ( Q Q mp n PmQ n < m K, Q ) n 1. K Q mp n P mq n R P n = P m. Q n Q m A(x) A(x) = P N(x) Q N (x) = A N(x), Q N (x) K = Q N (x) A N (x) = A M (x), A(x) A (x) < Q N (x) 2. A (x) A(x) A(x) = P (x)/q(x) P (x) Q(x) A(x) x d P (x) Q(x) A(x) Q(x) < d 2

[n, k, n k + 1] q q F q n = q 1 n < q k n n k + 1 = (m 0, m 1,..., m k 1 ) m i F q i = 0,..., k 1

m(x) = m 0 + m 1 x + + m k 1 x k 1 F n q m(x) n a 0, a 1,..., a n 1 F q = (c 0, c 1,..., c n 1 ) = (m(a 0 ), m(a 1 ),..., m(a n 1 )). a 0, a 1,..., a n 1 i = 0,..., n 1 a i = α i α F q α α 0,..., α n 1 F q C = { (m(α 0 ), m(α 1 ),..., m(α n 1 )) m(x) F q [x], m < k } 1 1 1... 1 = { Fq} k 1 α α 2... α n 1 = 1 α k 1 α 2(k 1)... α (n 1)(k 1). C [n, k, n k + 1] q 1 α α 2... α n 1 1 α 2 α 2 2... α 2(n 1) =. 1 α n k α 2(n k)... α (n k)(n 1) m 0,..., m k 1 C [n, k] T = k (n k). T (i, j) {0,..., k 1} {0,..., n k 1} n 1 n 1 ( T ) i,j = α il α l(j+1) = l=0 l=0 α l(i+j+1) n 1 = (α i+j+1 ) l = (αi+j+1 ) n 1 α i+j+1 l=0 }{{} 1 1 =1 = ( {}}{ α n ) i+j+1 1 α i+j+1 1 = 0. n n = (α ij ) n 1 i,j=0 n k n k n k + 1 n = (v ij ) n 1 i,j=0 = (aj i )n 1 i,j=0 a 0, a 1,..., a n 1 0 i j n 1 (a j a i ). a 0, a 1..., a n 1

C = (c 0, c 1,..., c n 1 ) F n q T = T c(x) = n 1 c ix i F q [x] n c(α i ) = 0 i = 1,..., n k C {c(x) ( F q [x] n c(α i = 0, i = 1,..., n k} n k ) C i=1 (x αi ). = (m 0,..., m k 1 ) m(x) = m 0 + m 1 x + + m k 1 x k 1 [n, k] q n k g(x) = (x α i ) = g 0 + g 1 x + + g n k 1 x n k 1 + x n k, i=1 α F q m(x) g(x). n m(x)g(x) = c(x) F q [x] n { } n 1 C = (c 0, c 1,..., c n 1 ) c i x i = c(x) = m(x)g(x), m(x) F q [x], m < k. C = C n k α g(x) = n k i=1 (x αj+i ) j N 0. j = 0

F k q = ( ) k k k (n k). k m(x) = m 0 + m 1 x + + m k 1 x k 1 m(x) g(x) c(x) = x n k m(x) p(x), p(x) = x n k m(x) g(x), p(x) n k 1 n k p(x) = p 0 + p 1 x + + p n k 1 x n k 1 c(x) = c 0 + c 1 x + + c n 1 x n 1 = (c 0, c 1,..., c n 1 ) = (p 0, p 1,..., p n k 1, m 0, m 1,..., m k 1 ). g(x) c(x) c(x) = x n k m(x) p(x) p(x) p(x) = 0 g(x). [6, 2, 5] 7 Z 7 g(x) = (x 3)(x 3 2 )(x 3 3 )(x 3 4 ) = x 4 + 6x 3 + 3x 2 + 2x + 4. = (1, 3) m(x) = 3x + 1 c(x) = x 4 (3x + 1) (2x 3 + 3x 2 + x + 5) = 3x 5 + x 4 + 5x 3 + 4x 2 + 6x + 2, = (2, 6, 4, 5, 1, 3)

[n, k, d] q g(x) = n k i=1 (x αi ), α F q t t = n k 2 c(x) = m(x)g(x) = c 0 + c 1 x + + c n 1 x n 1 e(x) = e 0 + e 1 x + + e n 1 x n 1 F q [x] r(x) = c(x) + e(x) = r 0 + r 1 x + + r n 1 x n 1 e i 0 0 i n 1 i ν 0 ν t j 1 < j 2 < < j ν r(x) α, α 2,..., α n k i = 1, 2,..., n k n 1 S i = r(α i ) = r j α ij j=0 i c(x) = m(x)g(x) g(α i ) = 0 i = 1, 2,..., n k S i = r(α i ) = c(α i ) + e(α i ) = m(α i )g(α i ) + e(α i ) n 1 = e(α i ) = e j α ij i = 1, 2,..., n k j=0 r(x) k = 1, 2,..., ν X k = α j k,

Y k = e jk, X k Y k ν ν S i = e jk α ij k = Y k Xk i i = 1, 2,..., n k k=1 k=1 n k 2ν ν Λ(x) = (1 xx k ) = 1 + Λ 1 x + + Λ ν x ν, k=1 X 1 1 k ν k Y k X j+ν k j 0 = Λ(X 1 k ) 0 = 1 + Λ 1 X 1 k + + Λ ν X ν k 0 = Y k X j+ν k + Λ 1 Y k X j+ν 1 k + + Λ ν Y k X j k. k = 1 ν ν 0 = (Y k X j+ν k + Λ 1 Y k X j+ν 1 k + + Λ ν Y k X j k ) 0 = k=1 ν k=1 Y k X j+ν k + Λ 1 ν k=1 S i = ν k=1 Y kx i k Y k X j+ν 1 k + + Λ ν S j+ν + Λ 1 S j+ν 1 + + Λ ν S j = 0 ν Y k X j k, k=1 1 j 2t ν Λ 1 S j+ν 1 + + Λ ν S j = S j+ν. ν 1 j ν ν ν

Ω(x) = S(x)Λ(x) x 2t, S(x) = S 1 + S 2 x + + S 2t x 2t 1 Λ(x) Ω(x) S(x) Ω(x) = S(x)Λ(x) ( 2t ν = = = = = i=1 k=1 x 2t (1 X j x) ) x 2t ( ν Y k Xkx i i 1) i=1 k=1 j=1 ( 2t ) ( ν Y k X k X i 1 k x i 1 (1 X k x) ) X j x) j k(1 ν 2t Y k X k (1 X k x) k=1 ν k=1 i=1 (X k x) i 1 j k (1 X j x) x 2t Y k X k (1 Xk 2t x 2t ) (1 X j x) x 2t j k ν Y k X k (1 X j x). k=1 j k x 2t Λ(x) = ν Ω(x) ν 1 Y k = Ω(X 1 k ) Λ (X 1 k ) k = 1, 2,..., ν, Λ (x) Λ(x) x Λ (x) = X k X j x) + (1 X k j k(1 x)( (1 X j x)) j k Ω(X 1 k ) Λ (X 1 k ) = Y kx k j k (1 X jx 1 k ) X k j k (1 X jx 1 k ) = Y k. S(x) = S 1 x +... S 2t x 2t Ω(x) = (1 + S(x))Λ(x) x 2t+1

c(x) = 3x 5 + x 4 + 5x 3 + 4x 2 + 6x + 2 e(x) = 4x + 2x 2 r(x) = c(x) + e(x) = 3x 5 + x 4 + 5x 3 + 6x 2 + 3x + 2. S 1 = r(3) = 2 S 2 = r(3 2 ) = 2 S 3 = 5 S 4 = 6 S 5 = 0 S 6 = 6 r(x) X 1 = 3 X 2 = 3 2 = 2 Y 1 = 4 Y 2 = 2 Λ(x) = (1 3x)(1 2x) = 6x 2 + 2x + 1 Ω(x) = 4 3 (1 2x) + 2 2 (1 3x) = 6x + 2. Λ(x) 5 1 = 3 = X 1 4 1 = 2 = 3 2 = X 2 Y 1 = Ω(3 1 ) Λ (3 1 ) = 6 3 1 + 2 3 (1 2 3 1 ) = 4 6 = 4, Y 2 = Ω(2 1 ) Λ (2 1 ) = 6 2 1 + 2 2 (1 3 2 1 ) = 5 1 = 2. e(x) = 4x + 2x 2, r(x) c(x) = r(x) e(x) = 3x 5 +x 4 +5x 3 +4x 2 +6x+2 m(x) = 3x + 4 Λ(x)

S 1, S 2,..., S 2t L N 0 C(x) = 1 + C 1 x + + C L x L s 1, s 2,..., s N N L s j = L C i s j i j = L + 1, L + 2,..., N. i=1 Λ(x) n n = 1 S 1, S 2,..., S n Λ (n) (x) = 1+Λ (n) 1 x+ +Λ (n) x L(n) L (n) L (n) Λ (n 1) (x) S 1, S 2,..., S n 1 S 1, S 2,..., S n D (n) = S n + L (n 1) j=1 Λ (n 1) j S n j. Λ (n) (x) = Λ (n 1) (x) L (n) = L (n 1) Λ (n) (x) n 1 Λ (n) (x) = Λ (n 1) (x) D (n) T (x), T (x) = x n u Λ (u 1) (x)/d (u), u L (n) = {L (n 1), n L (n 1) }. C L C(x) L

S 1, S 2,..., S 2t Λ(x) n 0 Λ (0) (x) 1 L (0) 0 T (x) x n < 2t n n + 1 D (n) S n + L (n 1) j=1 Λ (n 1) j S n j D (n) = 0 Λ (n) (x) Λ (n 1) (x) L (n) L (n 1) Λ (n) (x) Λ (n 1) (x) D (n) T (x) 2L (n 1) < n L (n) n L (n 1) T (x) Λ (n 1) (x)/d (n) L (n) L (n 1) T (x) x T (x) Λ(x) = Λ (2t) (x) Λ (n) (x) L (n) S 1, S 2,..., S n L (n) S j + Λ (n) i S j i = 0 j = L (n) + 1, L (n) + 2,..., n. i=1 Λ (n 1) (x) Λ (u 1) (x). Λ (u) (x) L (n) S j + Λ (n) i S j i i=1 = S j + = { L (n 1) i=1 Λ (n 1) i S j i D(n) D (u) S j (n u) + L (u 1) i=1 Λ (u 1) i S j (n u+i) 0 D(n) 0 = 0 D (u) j = L (n) + 1, L (n) + 2,..., n 1 D (n) D(n) D (u) = 0 D (u) j = n. Λ (n) (x) S 1, S 2,..., S n Λ (2t) (x) S 1, S 2,..., S 2t L (n) S 1, S 2,..., S n

S 1 = 2 S 2 = 2 S 3 = 5 S 4 = 6 S 5 = 0 S 6 = 6 n D (n) Λ (n) (x) L (n) T (x) 0 1 0 x 1 2 5x + 1 1 4x 2 5 6x + 1 1 4x 2 3 3 2x 2 + 6x + 1 2 2x 2 + 5x 4 5 6x 2 + 2x + 1 2 2x 3 + 5x 2 5 0 6x 2 + 2x + 1 2 2x 4 + 5x 3 6 0 6x 2 + 2x + 1 2 2x 5 + 5x 4 Λ(x) = Λ (6) (x) = 6x 2 +2x+1 Ω(x) = Λ(x) S(x) x 6 = (6x 2 + 2x + 1) (6x 5 + 0x 4 + 6x 3 + 5x 2 + 2x + 2) x 6 = 0x 5 + 0x 4 + 0x 3 + 0x 2 + 6x + 2 = 6x + 2. R η N r N = 0 r n n = 1, 0,..., N r n = u n a + v n b. u n+1 v n v n+1 u n = ( 1) n n = 1, 0,..., N u n v n

(a, b) a, b R η(a) η(b) u, v R ua + vb = (a, b) n 0 r 1 a u 1 1 v 1 0 r 0 b u 0 0 v 0 1 r n 0 n n + 1 q n, r n R r n 2 = q n r n 1 + r n η(r n ) < η(r n 1 ) u n u n 2 q n u n 1 v n v n 2 q n v n 1 (a, b) = r n 1 u = u n 1 v = v n 1 R F q [x] η q r a(x) a b(x) b S(x) Λ(x) Ω(x) Λ(x) t Ω(x) < t Λ(0) = 1 Ω(x) = S(x)Λ(x) x 2t. a(x) = x 2t b(x) = S(x) n = 1, 0,..., N r n (x) = u n (x) x 2t + v n (x) S(x), r n (x) = v n (x) S(x) x 2t. v n (x) r n (x) Λ(x) Ω(x) n = 1,..., N v n (x) = n q i (x) i=1 q n (x) = r n 2 (x) r n 1 (x).

r n 1 (x) = r n 2 (x) q n (x) = ( r n 3 (x) q n 1 (x)) q n (x) n = r 1 (x) q i (x) i=1 = a(x) v n (x). K > 0 r K (x) < t r K 1 (x) t, v K (x) = a(x) r K 1 (x) 2t t = t. v K (x) r K (x) v K (x) r K (x) v K (x) v K (0) 1 Λ(x) = v K (0) 1 v K (x), Ω(x) = v K (0) 1 r K (x). Λ(x) Ω(x) u n (x) S(x) = S 1 + S 2 x + + S 2t x 2t 1 Λ(x) Ω(x) n 0 r 1 (x) x 2t r 0 (x) S(x) v 1 (x) 0 v 0 (x) 1 r n (x) t n n + 1 q n (x) [r n 2 (x)/r n 1 (x)] r n (x) r n 2 (x) q n (x) r n 1 (x) v n (x) v n 2 (x) q n (x) v n 1 (x) Λ(x) = v n (0) 1 v n (x) Ω(x) = v n (0) 1 r n (x) v K (0) = 0 r K (0) = 0 x y [x/y]

S(x) = 2 + 2x + 5x 2 + 6x 3 + 0x 4 + 6x 5. n r n (x) v n (x) q n (x) 1 x 6 0 0 6x 5 + 0x 4 + 6x 3 + 5x 2 + 2x + 2 1 1 6x 4 + 5x 3 + 2x 2 + 2x + 0 x 6x 2 6x + 2 6x 2 + 2x + 1 x + 5 Λ(x) = v 2 (0) 1 v 2 (x) = 6x 2 + 2x + 1, Ω(x) = v 2 (0) 1 r 2 (x) = 6x + 2. 1 1 X k x = 1 + X kx + X 2 kx 2 +.... X k Y k k Y k X k k = 1 ν ν k=1 Y k X k 1 X k x = S 1 + S 2 x + S 3 x 2 +.... ν k=1 Y kx k j k (1 X jx) ν k=1 (1 X = S 1 + S 2 x + S 3 x 2 +..., kx) Ω(x) Λ(x) = S 1 + S 2 x + S 3 x 2 + + S 2t x 2t 1 +.... S (x) S (x) x x 1 S (x 1 ) F ((x))

S (x) = x 1 S (x 1 ) S (x) S (x) = Ω(x)/ Λ(x) Ω(x) Λ(x) F [x] Ω(x) = x ν 1 Ω(x 1 ), Λ(x) = x ν Λ(x 1 ). Λ(x) Λ(x) Ω(x) Ω(x) Ω(x) S (x) = Ω(x)/ Λ(x) Ω(x) Λ(x) = xν 1 Ω(x 1 ) x ν Λ(x 1 ) = x 1 S (x 1 ) = S (x). Λ(x) Ω(x) Λ(x) = x ν Λ(x 1 ) = x ν ν (1 x 1 X k ) = k=1 Ω(x) = x ν 1 Ω(x 1 ) = x ν 1 = ν Y k X k (x X j ). k=1 j k ν (x X k ), k=1 ν Y k X k (1 X j x 1 ) k=1 j k Λ(x) X k S (x) = Ω(x)/ Λ(x) Ω(x) Λ(x) S (x) 2t 2t 1 Ω(x) Λ(x) S (x) Λ(x) < 2t+1 2 Λ(x) = ν t < 2t+1 2 S (x) Ω(x) Λ(x) Ω(x) Λ(x) x Λ(x) λ Ω(x) Λ(x) S (x) Λ(x) = ν ν ν (x) = 0 ν t t

S (x) = S 1 x 1 + S 2 x 2 + + S 2t x 2t + X 2t 1 Λ(x) Ω(x) P 1 (x) 1 Q 1 (x) 0 1 (x) 1 P 0 (x) 0 Q 0 (x) 1 0 (x) S (x) n 0 n (x) n n + 1 [ ] a n (x) n 2 (x) n 1 (x) P n (x) a n (x)p n 2 (x) + P n 1 (x) Q n (x) a n (x)q n 2 (x) + Q n 1 (x) n (x) a n (x) n 2 (x) + n 1 (x) Λ(x) = x ν λq n (x 1 ) Ω(x) = x ν 1 λp n (x 1 ) ν = Q n (x) λ = 1 (Q n (x)) S (x) = 2x 1 + 2x 2 + 5x 3 + 6x 4 + 0x 5 + 6x 6 + X 7, X l l Z, x l n P n (x) Q n (x) n (x) a n (x) 1 1 0 1 0 0 1 2x 1 + 2x 2 + 5x 3 + 6x 4 + 0x 5 + 6x 6 + X 7 0 1 1 4x + 3 5x 2 + 4x 3 + 4x 4 + 3x 5 + X 6 4x + 3 2 x + 3 4x 2 + x + 3 X 5 x + 3 Λ(x) = x 2 4 1 (4x 2 + x 1 + 3) = 1 + 2x + 6x 2, Ω(x) = x 4 1 (x 1 + 3) = 2 + 6x.

p(x) = n p i x i = p(x) = n p n i x i = x n p(x 1 ). a(x) = n 1 a ix i b(x) = n 2 j=0 b jx j n a(x)b(x) = c(x) = c k x k, n = n 1 + n 2, d k = k=0 n = n 1 + n 2, c k = â(x) b(x) = d(x) = (n 1 i)+ +(n 2 j)=k i+j=k a i b j. n d k x k, k=0 a i b j = â(x) b(x) = ĉ(x) = i+j=n k a(x)b(x). a i b j = c n k.

x 2t Λ(x)S(x) = 2t+ν 1 k=0 ν 1 = (3.6) = k=0 ν 1 k=0 k Λ i S k+1 i x k k Λ i S k+1 i x k + 2t 1 k=ν k Λ i S k+1 i x k + 0 + = Ω(x) + x 2t A(x), k Λ i S k+1 i x k + 2t+ν 1 k=2t 2t+ν 1 k=2t k Λ i S k+1 i x k A(x) ν 1 k Λ i S k+1 i x k Λ(x)Ŝ(x) = Λ(x)S(x) = x 2t Ω(x) + Â(x). A(x) Ω(x) S(x) = Ω(x) + [ ] x2t A(x) x 2t A(x) =, Λ(x) Λ(x) Λ(x) = ν Ω(x) ν 1 [ ] [ Ŝ(x) = x2t Ω(x) + Â(x) x 2t Ω(x) ] =. Λ(x) Λ(x) A(x) S(x) = 2 + 2x + 5x 2 + 6x 3 + 0x 4 + 6x 5 Λ(x) = 1 + 2x + 6x 2. Λ(x)S(x) = 2 + 6x + 0x 2 + 0x 3 + 0x 4 + 0x 5 + 5x 6 + 1x 7, Ω(x) = 2 + 6x, A(x) = 5 + x. [ ] [ ] x 6 A(x) x 7 + 5x 6 = Λ(x) 6x 2 + 2x + 1 = [ 6x 5 + 0x 4 + 6x 3 + 5x 2 + 2x + 2 + 6x 1 + 0x 2 + 6x 3 +... ] = 6x 5 + 0x 4 + 6x 3 + 5x 2 + 2x + 2 = S(x). Ω(x) = x ν 1 Ω(x 1 ) Â(x) = x ν 1 A(x 1 ) ν 1

A(x) u n K u K (x)x 2t + v K (x)s(x) = r K (x) v n (0) 1 Λ(x) Ω(x) v K (0) 1 u K (x)x 2t + Λ(x)S(x) = Ω(x), A(x) = v K (0) 1 u K (x). Ŝ(x) K u K (x)x2t + v K (x)ŝ(x) = r K (x). λv K (x) = Λ(x) = x ν Λ(x 1 ) = x ν v K (0) 1 v K (x 1 ), λu K (x) = Ω(x) = x ν 1 Ω(x 1 ) = x ν 1 v K (0) 1 r K (x 1 ), λr K (x) = Â(x) = xν 1 A(x 1 ) = x ν 1 v K (0) 1 u K (x 1 ). λ = 1 (v K (x)) v K (x) Ŝ(x) = 2x5 +2x 4 +5x 3 +6x 2 +0x+6 u n (x) n r n (x) u n (x) v n (x) q n (x) 1 x 6 1 0 0 2x 5 + 2x 4 + 5x 3 + 6x 2 + 0x + 6 0 1 1 2x 4 + 3x 3 + 3x 2 + 4x + 3 1 3x + 4 4x + 3 2 6x + 4 6x + 4 4x 2 + 1x + 3 x + 3 S(x) Ŝ(x) S(x) Ŝ(x) f = P /Q F ((x)) P, Q F [x] P < Q f P Q

r n /Q = ( 1) n n u n = ( 1) n+1 P n v n = ( 1) n Q n n = 1, 0,..., N, a n = q n n = 1, 2,..., N, N n r 1 /Q = Q/Q = 1 = 1, r 0 /Q = P /Q = f = f [f] = 0, u 1 = 1 = P 1, u 0 = 0 = [f] = P 0, v 1 = 0 = Q 1, v 0 = 1 = Q 0. n 1 n 1 [ ] [ ] [ rn 2 ( 1) n 2 n 2 a n = = = ] n 2 = q r n 1 ( 1) n 1 n, n 1 n 1 r n = r n 2 q n r n 1 = ( 1) n 2 Q n 2 a n ( 1) n 1 Q n 1 = ( 1) n Q ( n 2 + a n n 1 ) = ( 1) n Q n. u n v n r n S (x) = Ω(x)/Λ(x) S (x) Ŝ(x)/x2t Ŝ(x)/x2t F ((x)) Ŝ(x) x2t Ŝ(x) x2t K Ŝ(x) Ŝ(x) S(x) Ŝ(x) S i i = 1 2t Ŝ(x)/x 2t x 2t 1 S (x) Ŝ(x)/x2t < 2 2t

r 1 (x) = a(x) + b(x) v 1 (x) = 1 r n 2 (x) = u n 2 (x)a(x) + v n 2 (x)b(x) C Cr n 2 (x) = Cu n 2 (x)a(x) + Cv n 2 (x)b(x). u n (x) = Cu n 2 (x) q n (x)u n 1 (x), v n (x) = Cv n 2 (x) q n (x)v n 1 (x). C r n 2 (x) r n 1 (x) D n 2 x δ D n 1 x γ D r n 2 (x) C q n (x) q n (1) (x) = CD n 2x δ D n 1 x. γ q n (x) = q n (1) (x) r n (1) (x) r n 1 (x) n + 1 r n (1) (x) q n (1) (x) (j 1) q n (x) r n (x) q n (j 1) (x) r n (j 1) (x) j r n (j 1) (x) r n 1 (x) D n x γ D n 1 x δ D n D n 1 x γ δ r n 1 (x) r n (j 1) (x) q n (j) (x) = q n (j 1) (x) + D n x γ δ, D n 1 δ γ

r n (j) (x) = Cr n 2 (x) q n (j) (x)r n 1 (x), r n (j) (x) r n (j 1) (x) q n (x) r n (x) a(x) = x 2t v n (x) a(x) = x 2t n 1 r n (x) 2t b(x) = Ŝ(x) r n (x) = v n (x)ŝ(x) x2t. L n v n (x) γ r n (x) L n j=0 (v n ) j S 2t (γ j), (v n ) j j v n (x) S 2t (γ j) γ j Ŝ(x) = S 2t + S 2t 1 x + + S 1 x 2t 1. r n (x) v n (x) Ŝ(x) L n j=0 (v n ) j S 2t (γ j), γ r n (x) γ i r n 1 (x) δ γ < δ r n (x) r n 1 (x) n v n (x) C q n (j) (x) q n (x) v n (j) (x) v n (x) v n (x) v n (x) v n 1 (x) T (x) r n δ v n (x) L n

Ŝ(x) = S 1 x 2t 1 + S 2 x 2t 2 + + S 2t Λ(x) C = D n 2 /D n 1 n 0 v 0 (x) 1 L 0 0 T (x) 1 i 0 v 1 (x) 1 δ 2t D 1 1 γ 6 i < 2t γ t i i + 1 γ 2t + L n i γ r n (x) = v n (x) S(x) D n = L n j=0 (v n) j S 2t (γ j) D n 0 γ < δ n n + 1 v n (x) C v n 2 (x) C (D n 2 /D n 1 ) x δ γ v n 1 (x) T (x) v n 1 (x)/d n 1 δ γ L n v n v n (x) v n (x) D n T (x) x γ δ Λ(x) = x v n(x) v n (x 1 ) C = D n 1 /D n 2 v n (x) = D n 1 T (x) x δ γ v n 1 (x) γ δ r n (x) r n 1 (x) r n 1 (x) r n (x) v n (x) v n (j) (x) = Cv n 2 (x) q n (j) (x) v n 1 (x)x γ δ. v n (x) q n (x) q n (j) (x) v n (j) (x) = v n (j 1) (x) D n T n 1 (x)x γ δ, T n 1 (x) = v n 1 (x)/d n 1 C = 1 C = D n 2 /D n 1 v 1 (x) = 1 v 0 (x) = 1 C Λ (i) (x) Λ (i) (x)

v n (x) v (1) n (x) = D n 1 T (x) x δ γ v n 1 (x). γ < δ γ = 2t + L n i δ = 2t + L n 1 i 0 L n = i 0 L n 1 2L n < i v n (x) i Λ (i) (x) D (n) D n T (x) x C D n 1 /D n 2 i = 0 γ = 6 D 1 = 1 n = 0 () = () = δ = 6 = i = 1 γ = 5 D 0 = 2 n = 1 () () = + () = δ = 5 = i = 2 γ = 5 = () () = + i = 3 γ = 4 D 1 = 4 n = 2 () () = + + () = + δ = 4 = i = 4 γ = 4 = () () = + + i = 5 γ = 3 D 2 = 0 i = 6 γ = 2 D 2 = 0 v (1) 1 (x) v (2) 1 (x) 6 1 v 1 (x) v 2 (x)