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1 This note is based on the popular mathematics mini-course that I gave at the department of mathematics of Kyoto University on August 6 8, The participants included high school students, first year university students and high school teachers The MacTutor History of Mathematics archive We believe that most of the images are in the public domain and that provided you use them on a website you are unlikely to encounter any difficulty Maple kawaguch(())math.kyoto-u.ac.jp

2 A (1.1) 36 B 38 x 2 f(x) = X x X x = 1 f 1 f f f f f x = 3 4 f 3 4 f 5 4 f 1 4 f 7 4 f 5 4 f 1 4 f f 1 4 f 7 4 f 5 4 f f f 3 4 f f A. (1) f(x) = X (2) c f(x) X 2 + c 1 3 c f(x) = X 2 + c A B. c 2 f(x) = X 2 + c A f

3 1 3 B c = f(x) = X f 7 4, 1 4, f 3 5 4, 3 4, 1 4, 3 4, c f(x) = X2 + c f 3 A f 4 6 B x x a b a, b b H(x) = max{ a, b} H(0) = max{0, 1} = 1 H(x) x height 1.1. (1) H(1) = max{1, 1} = 1. (2) H(0.999) = max{999, 1000} = (3) H( 0.255) = max{51, 200} = 200. H(x) 1.2. x (1) H(x) 1 (2) (1) x = 0, 1, 1 (3) B 1 B {x H(x) B}

4 1 4 (1) x a b b H(x) = max{ a, b} b 1 (2) H(x) = 1 b = 1 a b a a = 1, 0, 1 x = 1, 0, 1 (3) B = 3 H(x) 3 a 3 b 3 H(x) 3 x = a b a 3, 2, 1, 0, 1, 2, 3 7 b 1, 2, 3 3 H(x) = 21 a, b 21 B [B] B H(x) B [B],..., 0, 1,..., [B] 1, 2,..., [B] (2[B] + 1) [B] B N(B) # N(B) = #{x H(x) B} 1.2(1)(2) N(1) = 3 1.2(3) N(3) 21 N(3) x a b a, b H(x) 3 x N(3) = 15 3, 2, 1, 0, 1, 2, 3, 3 2, 1 2, 1 2, 3 2, 2 3, 1 3, 1 3, (1) B = 1, 2,..., 5 N(B) N(B) B B = 1, 3 2 B = 6,..., 10 B N(B) N(B)/B /9 = /36 = /49 = /64 = /81 = /100 = 1.27

5 1 5 (2) B N(B) N(B) B 2 B = 1, 2,..., 100 N(B) B ( B, N(B) B 2 ) B = 1, 2,..., N(1), N(2) 4, N(3) 9,..., N(B) B 2,... B (1.1) N(B) lim B B 2 = 12 π 2 = (1.1) A π H(x) 1.4. (1) x H( x) = H(x) (2) x H(x 2 ) = H(x) 2 (1) x = 0 H(0) = H(0) x 0 (1) x = a a b a, b b x x = b H( x) = max{ a, b} = max{ a, b} = H(x) (2) (1) x 0 (1) x = a b a, b b x 2 x 2 = a2 b a, b a 2, b 2 2 H(x 2 ) = max{a 2, b 2 } = (max{ a, b}) 2 = H(x) 2

6 1 6 H(x) (1.2) h(x) = log H(x) (= log max{ a, b}) h(x) x *1 h(0) = 0, h(1) = 0, h(0.999) = log 1000, h( 0.255) = log x a b a 1, 000 b 10, 000 b h(x) log b log(10 10,000 ) log 10 = h(x) 10, = 23, 000 h(x) x h(x) 1.5. x (1) h(x) 0 (2) (1) x = 0, 1, 1 (3) T 0 T {x h(x) T } (4) h( x) = h(x) (5) h(x 2 ) = 2h(x) 1.6. (1) n x h(x n ) = nh(x) (2) x, y h(x + y) max{h(x), h(y)} + log 2 H(x) h(x) h(x) 1.7. a 0 X d + a 1 X d a d (a 0,..., a d a 0 0) α 3 X 2 3 = 0 α p(x) = a 0 X n + a 1 X n a n (a 0,..., a n a 0 0) *1 h(x) x logarithmic height

7 1 7 p(α) = 0 p(x) a 0,..., a n a 0,..., a n a 0,..., a n a 0,..., a n 1 p(x) = a 0 X n + a 1 X n a n a 0,..., a n 1 n a 0 > 0 α p(x) p(x) = 0 α 1, α 2,..., α n α h(α) ( ) (1.3) h(α) = 1 n log a 0 + log max{1, α i } n i=1 α α = a b a, b b α p(x) bx a bx a 1 (1.3) a b ( a ) { h = log b + log max 1, a } = log max{b, a } b b (1.2) 3 p(x) X 2 3 p(x) = 0 3, 3 ( ) h 3 = 1 ( { log 1 + log max 1, } 3 2 { + log max 1, }) 3 = 1 2 log x x x = (x 1,..., x n ) i = 1,..., n x i ai b i d b 1,..., b n d b i d i (1.4) x i = a i = a id i b i d x 1,..., x n d x i = a id i d x = (x 1,..., x n ) H(x) (1.5) H(x) = max{ a 1 d 1,..., a n d n, d} n = 1 H(x) ( 1 4, 5 6, 5) 2 4, 6, 5 60 ( 1 4, 5 6, 2 ) 5 = ( 15 60, 50 60, 24 ) 60 (( 1 H 4, 5 6, 2 )) = max { 15, 50, 24, 60} = 60 5

8 n (1) x = (x 1,..., x n ) H(x) 1 (2) (1) i = 1,..., n x i 1, 0, 1 (3) B 1 B x = (x 1,..., x n ) {x = (x 1,..., x n ) Q n H(x) B} (1) x = (x 1,..., x n ) (1.4) d H(x) = max{ a 1 d 1,..., a n d n, d} d 1 (2) H(x) = 1 d = 1 a i d i 1 i = 1,..., n d i d d = 1 a i = 1, 0, 1 i = 1,..., n i = 1,..., n x i 1, 0, 1 (3) B = 3 H(x) 3 d 3 a i d i 3 i = 1,..., n x i ai b i, a i a i d i 3 b i d 3 x i = ai b i = aidi d ai b i a i 3, 2, 1, 0, 1, 2, 3 7 b i 1, 2, 3 x i 7 3 = 21 i 1,..., n n H(x) 3 (21) n (a i, b i b 1,..., b n d 3 (21) n.) B H(x) B ((2[B] + 1) [B]) n (1.1) H(x) B B A (1.6) X 2 = 2Y 2 (0, 0) 2

9 1 9 (0, 0) (x 1, x 2 ) (1.7) x 2 1 = 2x 2 2 x 1, x 2 0 (x 1, x 2 ) x 2 (x 1, x 2 ) x 2 1 x 1 x 1 = 2x 3 x 3 (1.6) 2 (1.8) x 2 2 = 2x 3 0 (x 2, x 3 ), (1.6) (1.7) x 1 > x 2 (1.8) x 2 > x 3 (x 1, x 2 ) (x 2, x 3 ) (x 2, x 3 ) (x 1, x 2 ) (x 3, x 4 ) 0 (1.6) (1.6) (x 1, x 2 ) (x 2, x 3 )..., (x i, x i+1 )... x 1 > x 2 > > x i+1 x 1,..., x i+1 (x i, x i+1 ) (x i, x i+1 ) 0 (1.5) H ((x i, x i+1 )) = max{ x i, x i+1, 1} = max{ x i, x i+1 } (x 1, x 2 ) B x 1 > x 2 > > x i+1 H ((x i, x i+1 )) H ((x 1, x 2 )) = B i = 1, 2,... {y Q 2 H(y) B} i (x i, x i+1 ) 1.8(3) 1.9 X 2 = 2Y 2 (0, 0) (0, 0) X 2 = 2Y 2 descent B 1.8(3) size descent size 17 6

10 (1) (1.9) X 4 + Y 4 = Z 2 (x, y, z) x = 0 y = 0 (2) (a, b) b 2 = a (a, b) = (0, ±1) * (1) (1.9) (x, y, z) x 0, y 0 (x, y, z) (1.9) (r, s, t) r 0 s 0. (r, s, t) (x, y, z) (r, s, t) (x, y, z) (1.9) 1.8(3) x > 0, y > 0, z > 0 x y x y x 4 + y z 2 4 x, y y x y x 1.11 a > 0, b > 0, (1.10) x 2 = a 2 b 2, y 2 = 2ab, z = a 2 + b 2 a b x b x 2 + b 2 = a p > 0, q > 0, (1.11) x = p 2 q 2, b = 2pq, a = p 2 + q 2 a p q *2 L. Mordell, Diophantine equations, Academic Press, 1969.

11 1 11 y 2 = 2ab = 4pq(p 2 + q 2 ) p, q, p 2 + q 2 2 r > 0, s > 0, t > 0 (1.12) p = r 2, q = s 2, p 2 + q 2 = t 2 p q r s r 4 + s 4 = t 2 (1.9) (x, y, z) (1.9) (r, s, t) (r, s, t) (x, y, z) (r, s, t) (x, y, z) (1.10) (1.11) (1.12) z = a 2 + b 2 = (p 2 + q 2 ) 2 + (2pq) 2 = t 4 + (2r 2 s 2 ) 2 > t 4 z > t (r, s, t) (x, y, z) x 4 + y 4 = z 2 x, y, z z x, y, z H ((x, y, z)) = max{x, y, z, 1} = z H ((r, s, t)) = t z > t (r, s, t) (x, y, z) (1.9) (x, y, z) x 0, y 0 (1.9) (r, s, t) r 0, s 0 (r, s, t) (x, y, z) (1.9) 1.8(3) (1) (2) a = x y, z b = z y 2 x, y, z x 4 + y 4 = z 2 (1) x = 0 (a, b) = (0, ±1) x > 0, y > 0, z > 0 x 2 + y 2 = z 2 x, y, x y a > 0, b > 0 x = a 2 b 2, y = 2ab, z = a 2 + b 2 y 2 = z 2 x 2 = (z + x)(z x) z x z + x s, t z + x = 2s, z x = 2t d s, t d z = s + t x = s t x y x z d = 1 s t y 2 = 4st s t a > 0, b > 0, s = a 2, t = b 2

12 2 12 a, b x = a 2 b 2, y = 2ab, z = a 2 + b x = a b h(x) = log max{ a, b} 1.2 c 2 f c (X) = X 2 + c 2 2 f c (X) x x 2 f c f c (x) x h(x) f c (x) h(f c (x)) x h(x) h(f c (x)) 2.1. f 0 (X) = X 2 1.5(5) x (2.1) h(f 0 (x)) = 2h(x) f 0 (x) x f 1 (X) = X x a b ( a ) 2 a 2 + b 2 f 1 (x) = + 1 = b a b a 2 + b 2 b 2 f 1 (x) a2 +b 2 b 2 h(f 1 (x)) = log max{a 2 + b 2, b 2 } = log(a 2 + b 2 ). h(x) = log max{ a, b} (max{ a, b}) 2 = max{a 2, b 2 } max{a 2, b 2 } a 2 + b 2 2 max{a 2, b 2 } b 2 2h(x) h(f 1 (x)) log 2 + 2h(x) x h (f 1 (x)) 2h(x) log 2 f 1 (x) x 2 log 2

13 2 13 f c (X) = X 2 + c 2 2 x f c (x) x 2 c 2.3. c f c (X) = X 2 + c 2 c C x (2.2) h (f c (x)) 2h(x) C c C 1 x (2.3) h (f c (x)) 2h(x) + C 1 c C 2 x (2.4) 2h(x) h (f c (x)) + C 2 C C 1, C 2 (2.3) (2.4) (2.2) c = p q p, q q c x a b ( a ) 2 p fc(x) = + b q = a2 q + b 2 p b 2. q (2.5) h(f c (x)) max log { a 2 q + b 2 p, b 2 q } a2 q+b 2 p b 2 q (2.6) log max { a 2 q + b 2 p, b 2 q } log max {( a 2 q + b 2 p ), b 2 q } log max{a 2, b 2 } + log( p + q) + log 2 = 2 log max{ a, b} + log( p + q) + log 2 C 1 = log( p + q) + log 2 c log max{ a, b} x h(x) (2.5) (2.6) x h (f c (x)) 2h(x) + C 1,

14 2 14 (2.3) c = p q x = a b f c (x) = a2 q + b 2 p b 2 q h(f c (x)) log max{ a, b} log max{ a 2 q + b 2 p, b 2 q} log max{ a 2 q+b 2 p, b 2 q} log max{ a, b} h(f c (x)) log max{ a 2 q+b 2 p, b 2 q} a 2 q b 2 p qb 2 p +q a2 b 2 q a 2 q b 2 p a 2 q q p + q a2 p = q2 p + q a2 { } q max{ a 2 q + b 2 p, b 2 q} max{a 2 q b 2 p, b 2 2 q} max p + q a2, qb 2 a 2 q b 2 p qb 2 (2.7) q p +q a2 b 2 log max { a 2 q + b 2 p, b 2 q } { } q 2 log max p + q a2, b 2 q { } q log max{a 2, b 2 2 } + log min p + q, q { } q 2 = 2 log max{ a, b} + log min p + q, q b 2 q d b 2 = Bd q = q d B q ( a ) 2 p f c (x) = + b q = a2 q + Bp dbq m a 2 q + Bp dbq m dbq m 1, m 2, m 3 m = m 1 m 2 m 3 m 1, m 2, m 3 d, B, q, m 2 = 1 m 2 m a 2 q + Bp m 2 B m 2 a 2 q a b a 2 B m 2 B m 2 a 2 m 2 q B q m 2 = 1 m = m 1 m 3 dq = q f c (x) f c (x) = (a2 q + Bp)/m (dbq )/m

15 2 15 (dbq )d = b 2 q (a 2 q + Bp)d = a 2 q + b 2 p { a 2 q } + Bp (2.8) h(f c (x)) = log max, dbq m m = log max { a 2 q + Bp, dbq } log m = log max { a 2 q + b 2 p, b 2 q } log d log m log max { a 2 q + b 2 p, b 2 q } 2 log q { } q 2 C 2 = 2 log q log min p + q, q c log max{ a, b} x h(x) (2.7) (2.8) x 2h(x) h(f c (x)) + C 2, (2.4) d x 1.7 f(x) 2 2 d c 0,..., c d c 0 0 f(x) = c 0 X d + c 1 X d c d f C x (2.9) h (f(x)) dh(x) C [1, Chapter 4, Theorem 1.8] 2.5. x (2.9) 1 h (f(x)) h(x) d C d { 1 d n h (f n (x)) } f n (x) f(f( (f(x)) )) x f n 1 ĥ f (x) = lim n d n h (f n (x)) ĥf (x) x f ĥf

16 f(x) 2 x h(f(x)) = 2h(x) f(x) = X 2 f(x) = X f c (X) = X 2 + c f c (X) 2 2 f c (X) = X 2 + c 3.1 f(x) 2 2 d c 0,..., c d c 0 0 f(x) = c 0 X d + c 1 X d c d x x f iteration f n (x) f(f( (f(x)) )) x f n f 1 (x) = f(x) f 2 (x) = f(f(x)) f 3 (x) = f(f(f(x))) (3.1) x, f(x), f 2 (x), f 3 (x),... f x x x {f i (x )} {f i (x)} x Fatou set x Julia set f(x) = X 2 x x < 1 x x x < 1 {f i (x)} {f i (x )} 0 x x > 1 x x x > 1 {f i (x)} {f i (x )} x x = 1 {f i (x)} f i (x) = 1 x x x > 1 {f i (x )} x = 1 x x {f i (x)} {f i (x )} f(x) = X 2 J = {x C x = 1}

17 f(x) = X 2 {x C x = 1}. f(x) = X 2 f 4 f(x) = X 2 1. f (3.1) x 3.1. f(x) 2 x (1) x f-f-periodic point n f n (x) = x f n (x) = x n x prime period *3 *3 exact period minimal period

18 f(x) = X 2 + i. (2) x f- f-preperiodic point m f m (x) f- f- f-x n f- f- m n x f- f f f 0 (X) = X 2 f 1 (X) = X f 0-3.2(1) f f(x) = X 2 (1) f(0) = 0 0 f-0 1 (2) f(1) = 1 1 f-1 1 (3) ω = ω 1 3 f(ω) = ω 2 f 2 (ω) = f(ω 2 ) = ω 4 = ω ω f-f(ω) ω f 2 (ω) = ω ω 2 (4) i = 1 f(i) = i 2 = 1, f 2 (i) = f( 1) = 1 (2) 1 f- i f- (5) η = η 1 6 f(η) = ω (3) ω f- η f f(x) = X 2 n n f- 3.2(1)(2) n = 1 3.2(3) n = 2

19 3 19 f(x) f(x) 3.3 *4 3.4 (). f(x) 2 f(x) ax 2 (2b+1)X + (ab 2 + 2b) a 0 b n n f f(x) = ax 2 (2b + 1)X + (ab 2 + 2b) 2 f- 2 n n f- x n (f n ) (x) > 1 x repelling periodic point *5 z f z f. f(x) = X 2 f n (X) = X 2n f(x) = X 2 x = 0 x = cos ( ) 2πk 2 n + ( ) 2πk 1 sin 1 2 n 1 (n 1; 1 k 2 n 1) x = 0 P P {z C z = 1} (f n ) (X) = (2 n )X 2n 1 (f n ) (0) = 0 x = 0 P x (f n ) (x) = 2 n x 2n 1 = 2 n > 1 f(x) = X 2 P x =. z z = 1 z ε z P ε z z = 1, P z P {z C z = 1} f(x) = X 2 *4 I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), *5 x O(x) = {f n (x) n 1} x! d f(x) (2d 2). f(x).

20 f- f-rational f-periodic point f- f- rational f-preperiodic point 3.6. c 2 f c (X) = X 2 + c f c - {x Q x f c - } f c - f c c f C 2.3 x (3.2) h (f(x)) 2h(x) C x f-h(x) < 2C (3.3) {x Q x f- } {x Q h(x) < 2C} (3.3) 2C 1.5(3) f- (3.3) x h(x) 2C n 1 (3.4) h(f n (x)) (2 n + 1) C n = 1 (3.2) h(f(x)) 2h(x) C 3C n = k (3.4) x f k (x) (3.2) x f k (x) h(f k+1 (x)) 2h(f k (x)) C h(f k (x)) h(f k+1 (x)) 2h(f k (x)) C 2 ( 2 k + 1 ) C C = ( 2 k ) C n = k + 1 (3.4) (3.4) (3.3) x f- i, j 1 f i (x) = f i+j (x) f i (x) = f i+j (x) = f i+2j (x) = f i+3j (x) = (3.5) h(f i (x)) = h(f i+j (x)) = h(f i+2j (x)) = h(f i+3j (x)) = (3.4) (3.5) x f-h(x) < 2C (3.3)

21 d d d 2 f(x) d f- {x Q x f- } f d α 3.6 c 2 f c (X) = X 2 + c n { x C x n f c - } n f c - f c (X) d ( 2) f(x) α f(x) 2 α 1.7 p(x) p(α) = 0 p(x) = 0 G(α) G(α) = {β C P (β) = 0}. α α ĥ(α) 2.5 G(α) Autissier *6, Baker Hsia *7, Chambert-Loir, Baker Rumely, Favre Rivera-Letelier, Pineiro Szpiro Tucker *6 P. Autissier, Points entiers sur les surfaces arithmétiques, J. Reine Angew. Math. 531 (2001), *7 M. Baker and L.-C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics, J. Reine Angew. Math. 585 (2005),

22 f c (X) = X 2 + c c f c - f c - c f c (X) f c - f c c = 1 f 1 (X) = X , , 1 2 f f f 1-0, 1 f 1-0, 1, c = , , 5 4 f 21 (X) = X , f f , , , , 7 4, 1 4, 5 4 f 21 (X) c = f 29 (X) = X , 7 4, f 29-16

23 c = f c , 2 2 f c c = f c- c c 4 f c - f c - 5 f c - 9 *8, *9 4.5 (, ). c 2 f c (X) = X 2 + c x f c -x (). 4.5 f c - 4 f c c * 10 c c f c (X) = X 2 + c c x f c -x d g(x) = X d + c 1 X d c d (c 1,..., c d ) x g(x) = 0 x x a b a, b b g(x) = 0 b n 1 a n a n 1 a n 2 b n + c 1 b n 1 + c 2 b n 2 + c d = 0 a n b = c 1a n 1 c 2 a n 2 b c d b n 1 b = 1 x *8 P. Morton, J. H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 2 (1994), *9 B. Poonen, The classification of preperiodic points of quadratic polynomials over Q: a refined conjecture, Math. Z. 228 (1998), *10 R. Benedetto, Preperiodic points of polynomials over global fields, J. Reine Angew. Math. 608 (2007),

24 Narkiewicz * 11 x f c - n n = 1 n = 2 n 3 x g n (X) = f n c (X) X c g n (X) 1 g 2 (X) = f c (f c (X)) X = (X 2 + c) 2 + c X = X 4 + 2cX 2 X + c 2 + c g 2 (X) 1 4 x n g n (x) = x x 0, x 1,..., x n 1 (4.1) x 0 = x, x 1 = f c (x 0 ), x 2 = f c (x 1 ),..., x n 1 = f c (x n 2 ) x f c (X) x 0, x 1,..., x n 1 x n x 0, x 1,..., x n 1 x 0 = f c (x n 1 ) x n = x 0, x n+1 = x 1 x, y f c (y) f c (x) y x = y + x x y x i x i+1 f c (x i ) = x i+1 f c (x i+1 ) = x i+2 (4.2) x i+2 x i+1 x i+1 x i = x i + x i+1 i = 0, 1,..., n 1 (4.3) x 2 x 1 x 1 x 0 x 3 x 2 x 2 x 1 xn+1 x n x n x n 1 = (x 0 + x 1 )(x 1 + x 2 ) (x n 1 + x n ) x n = x 0, x n+1 = x 1 1 (x 0 + x 1 )(x 1 + x 2 ) (x n 1 + x 0 ) = 1. x 0,..., x n 1 i = 0, 1,..., n 1 x i + x i+1 1 1, i x i + x i+1 = 1 (4.2) x i+2 = x i n 3 x 0,..., x n 1 i x i + x i+1 = 1 x 0 + x 1 = 1 x 1 + x 2 = 1 x 0 = x 2 x 0,..., x n 1 n 3 n *11 W. Narkiewicz, Polynomial cycles in algebraic number fields, Colloq. Math. 58 (1989),

25 d 2 1 d f(x) = X d + c 1 X d c d c 1,..., c d f n n f c - n = 4, 5 n f c - (n 6.) c 2 f c (X) = X 2 + c x f c - (1) * 12 x 4 (2) * 13 x f c (X) 4 5 f 4 c (X) f 2 c (X) = f c (f c (X)) = (X 2 + c) 2 + c = X 4 + 2cX 2 + c 2 + c f 4 c (X) = f 2 c (f 2 c (X)) = (f 2 c (X)) 4 + 2c(f 2 c (X)) 2 + c 2 + c = X cX 14 + ( 28c 2 + 4c ) X 12 + ( 56c c 2) X 10 + ( 70c c 3 + 6c 2 + 2c ) X 8 + ( 56c c c 3 + 8c 2) X 6 + ( 28c c c c 3 + 4c 2 + ) X 4 + ( 8c c c c 4 + 8c 3) X 2 + ( c 8 + 4c c 6 + 6c 5 + 5c 4 + 2c 3 + c 2 + c ) f 4 c (X) X (4.4) f 4 c (X) X = ( f 2 c (X) X ) Φ c (X) Φ c (X) X 12 (4.5) Φ c (X) := X cX 10 + X 9 + ( 15c 2 + 3c ) X 8 + 4cX 7 + ( 20c c ) X 6 + ( 6c 2 + 2c ) X 5 + ( 15c c 3 + 3c 2 + 4c ) X 4 + ( 4c 3 + 4c ) X 3 + ( 6c c 4 + 6c 3 + 5c 2 + c ) X 2 + ( c 4 + 2c 3 + c 2 + 2c ) X + ( c 6 + 3c 5 + 3c 4 + 3c 3 + 2c ). 4 f c - x fc 4 (x) = x (4.4) fc 2 (x) x = 0 Φ c (x) = 0 fc 2 (x) = x x 2 x 4 Φ c (x) = 0 4 f c - Φ c (X) = 0 *12 P. Morton, Arithmetic properties of periodic points of quadratic maps. II., Acta Arith. 87 (1998), *13 E. V. Flynn, B. Poonen and E. F. Schaefer, Cycles of quadratic polynomials and rational points on a genus-2 curve, Duke Math. J. 90 (1997),

26 y x 6 Φ(X, Y ) = 0 xy Maple. 4.10(1) (x, y) x, y 5 f 5 c (X) X f 4 c (X) X = (f c (X) X) Ψ c (X) Ψ c (X) X 30 5 f c - Ψ c (X) = 0 30 Ψ c (X) Φ c (X) = 0 c x Ψ c (X) = 0 c x 4.10 Φ c (X) c Y Φ(X, Y ) Ψ c (X) c Y Ψ(X, Y ) (4.5) (4.6) Φ(X, Y ) = X Y X 10 + X 9 + ( 15Y 2 + 3Y ) X 8 + 4Y X 7 + ( 20Y Y ) X 6 + ( 6Y 2 + 2Y ) X 5 + ( 15Y Y 3 + 3Y 2 + 4Y ) X 4 + ( 4Y 3 + 4Y ) X 3 + ( 6Y Y 4 + 6Y 3 + 5Y 2 + Y ) X 2 + ( Y 4 + 2Y 3 + Y 2 + 2Y ) X + ( Y 6 + 3Y 5 + 3Y 4 + 3Y 3 + 2Y ). 2 X, Y Ψ(X, Y ) 2 X, Y Φ c (X) = 0 Ψ c (X) = 0 c x Φ(X, Y ) Ψ(X, Y ) (x, y) * 14 3 *14

27 Φ(X, Y ) = 0 Ψ(X, Y ) = F (X, Y ) 2 X, Y 4 Φ(X, Y ) Ψ(X, Y ) F (x, y) = 0 x, y (x, y) F (X, Y ) (5.1) F (X, Y ) = X + Y 1 F (X, Y ) = 0 t (x, y) = (t, 1 t) y x X + Y 1 = 0 xy t (t, 1 t). F (x, y) = (5.2) F (X, Y ) = X 2 + Y 2 1 ( ) 2t F (X, Y ) = 0. t (x, y) = 1+t, 1 t2 2 1+t 2 F (x, y) = 0 (x, y) = ( 3 5, ) ( 4 5, 5 12, ) ( 11 12, 8 17, 17) 15,... F (x, y) = 0

28 y x ( ) X 2 + Y 2 2t 1 = 0 xy t, 1 t2. 1+t 2 1+t x, y x 2 + y 2 = 1 (x, y) = ( 1, 0) ( ) 2t t (x, y) = 1+t, 1 t2 2 1+t (5.3) F (X, Y ) = X 3 Y 2 2 F (X, Y ) = 0 30 y x X 3 Y 2 2 = 0 xy (3, 5) ( 129 ( , ) ,...., ), (x, y) F (X, Y ) = 0 (5.4) ( x x 4y 2, x6 + 40x 3 ) y 3 F (X, Y ) = (x, y) = (3, 5) F (X, Y ) = 0, (x 1, y 1 ) = (3, 5)

29 5 29 (5.4) (x, y) (x n, y n ) (x n+1, y n+1 ) ( 129 (x 2, y 2 ) = 100, 383 ), 1000 ( (x 3, y 3 ) = , ) (x 4, y 4 ) y , (x 5, y 5 ) y (x 6, y 6 ) y (5.4) (x n, y n ) h(x) x (x n, y n ) (x n+1, y n+1 ) (x n, y n ) { 1 4 n 1 h(x n, y n ) } x. (x, y) = (3, 5) (5.4) F (X, Y ) = X 3 Y 2 2 = 0 (5.1) (5.2) (5.3) 5.5. P = (a, b) b 0 E : Y 2 = X 3 2 (1) E P l (2) E l P Q l P Q = P Q ( a a 4b 2, a6 + 40a 3 ) b 3, 2 F (X, Y ) F (X, Y ) = 0 genus 0 X + Y 1 = 0 X 2 + Y 2 1 = 0 0 X 3 Y = 0 1 F (X, Y ) = 0 1 F (X, Y ) = 0 2 F (x, y) = 0 x, y 5.6 (). F (X, Y ) 2 X, Y F (X, Y ) = 0 2 F (x, y) = 0 x, y

30 Φ(X, Y ) Ψ(X, Y ) Φ(X, Y ) = 0 2 Ψ(X, Y ) = 0 14 Φ(x, y) = 0 Ψ(x, y) = 0 (x, y) 4 5 f c - c c c ( 10 ). N F (X 1,..., X N ) F (X 1,..., X N ) = N F (X 1,..., X N ) F (X 1,..., X N ) = 0 YES NO 1960 F (X 1,..., X N ) 1970

31 F (X 1,..., X N ) = 0 N N = 2 2 F (X 1, X 2 ) F (X 1, X 2 ) = 0 F (X 1, X 2 ) = 0 2 F (X 1, X 2 ) 2 F (X 1, X 2 ) F (X 1, X 2 ) = 0 N = (1) N = 1 1 f(x) f(x) = 0 f(x) = c 0 X d + c 1 X d c d (c 0,..., c d ) f(x) = 0 (2) 1 f(x) f(x) f(x) f(x) = g(x)h(x) g(x), h(x) g(x) = ±1 h(x) = ±1 * 15 *15 III 4(2).

32 n 3 x n + y n = z n x, y, z 1 z 2 F (X, Y ) = X n + Y n 1 F (x, y) = 0 x = 0 y = 0 2 y x 1 12 X 3 + Y 3 = 1 xy (1, 0), (0, 1) ( ). 2 * 16 *16!

33 y x X 4 + Y 4 = 1 xy (±1, 0) (0, ±1) f c (1)

34 Φ(X, Y ) (4.6) X, Y X, Y (6.5) X = U 1 2(U + 1) + V 2U(U 1) Y = U 6 3U 5 3U 4 10U U 2 3U 1 4U(U 1) 2 (U + 1) 2 U, V Φ(X, Y ) X, Y U, V Φ(X, Y ) Θ(U, V ) = V 2 U(U 2 + 1)(1 + 2U U 2 ) Φ(X, Y ) Θ(U, V ) C Φ D Φ C Φ = {(x, y) C 2 Φ(x, y) = 0}, D Φ = {(u, v) C 2 Θ(u, v) = 0}. (x, y) (u, v) P 1,, P n C Φ Q 1,, Q m D Φ π : D Φ \ {Q 1,..., Q m } C Φ \ {P 1,..., P n }, (u, v) (x, y) π D Φ {Q 1,..., Q m } u = 0, 1, 1 (6.5) x, y π D Φ \ {Q 1,..., Q m } U, V C Φ \ {P 1,..., P n } π 1 1 C Φ D Φ D Φ 6.1 D Φ (0, 0), (±1, ±2) 5 5 D Φ {Q 1,..., Q m }, C Φ, C Φ \ {P 1,..., P n } C Φ {P 1,..., P n } C Φ 6.1. V 2 U(U + 1)(1 + 2U U 2 ) = 0 (u, v) (0, 0) (±1, ±2) 4 x y x = y 2 x square (u, v) v 2 = u(u + 1)(1 + 2u u 2 ) u, v u, u 2 + 1, 1 + 2u u 2 u u + 1, 1 + 2u u 2 2

35 u = a 2 u = b 2 a, b b 2 = a (2) (a, b) = (0, ±1) (u, v) = (0, 0) u = a 2 u = 2b 2 a, b 2b 2 = a (3) (a, b) = (±1, ±1) (u, v) = (±1, ±2) 6.2. (1) X 4 + Y 4 = 2Z 2 (x, y, z) x y (x, y) = (±1, ±1) (2) (a, b) 2b 2 = a (a, b) = (±1, ±1) f c (2) 4.10 Ψ c (X) fc 5 (X) X = (f c (X) X) Ψ c (X) x Ψ c (x) = 0 x 5 f c - 2 Ψ(X, Y ) Ψ c (X) c Y Ψ(X, Y ) (x, y) Ψ(X, Y ) X 30 Y 15 Ψ(X, Y ) = 0 x 5 f c - f c (x) 5 f c -, x Ψ c (x) = 0 Ψ c (f c (x)) = 0 Ψ(X, Y ) C Ψ C Ψ = {(x, y) C 2 Ψ(x, y) = 0}. (x, y) C 2 C Ψ (x 2 + y, y) C 2 C Ψ σ σ : C Ψ C Ψ, (x, y) (x 2 + y, y) x 5 f c -fc 5 (x) = x σ 5 : C Ψ C Ψ σ 5 (x, y) = (x, y) C Ψ P = (x, y) P, σ(p ), σ 2 (P ), σ 3 (P ), σ 4 (P ) 5 D Ψ π : C Ψ D Ψ

36 A (1.1) 36 P = (x, y) C Ψ π(p ) = π(σ(p )) = π(σ 2 (P )) = π(σ 3 (P )) = π(σ 4 (P )) Q D Ψ P C Ψ π 1 (Q) = {P, σ(p ), σ 2 (P ), σ 3 (P ), σ 4 (P )} π D Ψ, D Ψ (6.6) D Ψ : Y 2 = X 6 + 8X X X 3 + 5X 2 + 6X + 1 P = (x, y) C Ψ π(q) D Ψ C Ψ D Ψ C Ψ D Ψ 6.3. (6.6) (x, y) (0, 1), (0, 1), ( 3, 1), ( 3, 1), 4 6.3, Chabauty C Ψ P = (x, y) π(p ) D Ψ 4 2 π 1 ((0, 1)) C Ψ C Ψ P = (x, y) A (1.1) (1.1) k=1 1 k 2 = p: (1 1p 2 ) 1 = π2 6 p 1 (e = 0) µ(p e ) = 1 (e = 1) 0 (e 2)

37 A (1.1) 37 a a a = p e1 1 pe k k e 1,..., e k µ(a) = µ(p e 1 1 ) µ(pe k 1 ) p 1,..., p k µ(1) = 1 µ B, { M(B) = # (i, j) } i j 1 i, j B M(B) = #{x Q x H(x) B} 0 N(B) = 2M(B) + 1 B N(B) M(B) d { M d (B) = # (i, j) } i j d 1 i, j B [ B d ] B d i d, 2d,..., [ B d ]d [ B d ], i d, 2d,..., [ B d ]d [ B d ] M d(b) = [ ] B 2 d B B [ ] 2 B M(B) = µ(d)m d (B) = µ(d) d d=1 d=1 M(B) B 2 = B B d µ(d)[ B 2 = d=1 ] 2 B µ(d) 1 d 2 + R(B) d=1 R(B) B d=1 µ(d) 1 B 1 2 B d=1 µ(d) d 2 = p: e=0 N(B) = 2M(B) + 1 (1.1) µ(p e ) (p e ) 2 = p: M(B) lim B B 2 = 6 π 2 N(B) lim B B 2 = 12 π 2 (1 1p ) ( 2 = k=1 ) 1 1 k 2 = 6 π 2

38 B 38 A , n x = (x 1,..., x n ), N n (B) = #{x = (x 1,..., x n ) Q n H(x) B} n = 1 N(B) (A.1) N n (B) lim B B n+1 = 2 n ζ Q (n + 1) ζ Q (n + 1) = 1 k=1 k (A.1) (1.1) n+1 B 1.3. (1) B = 2, 4, 5 N(2) = 7, N(2)/2 2 = 7/4 = 1.75, N(4) = 23, N(2)/4 2 = 23/16 = , N(5) = 39, N(5)/5 2 = 39/25 = (2) C #include <stdio.h> /* */ int gcd(int x, int y) { if (y==0) return x; else return gcd(y, x%y); } int main() { int i, j, count, B, ans; printf(" B "); scanf("%d", &B); /* 1 <= i,j <= B gcd(i,j)=1 */ count = 0; for (i = 1; i <= B; i++) { for (j =1; j <= B; j++) { if (gcd(i,j) == 1) count++; } } /* 0, count 2 1 */

39 B 39 ans = 2 * count + 1; } printf("%d\n", ans); return 0; 1.6. (1) 1.4(2) (2) a, b, c, d max{ ad + bc, bd } max{2 ad, 2 bc, bd } 2 max{ a, b } max{ c, d }, x = a b, y = c d x + y = ad+bc ad h(x + y) log max{ ad + bc, bd } log max{ a, b } + log max{ c, d } + log 2 = h(x) + h(y) + log 2 h(x + y) log max{ ad + bc, bd } ad+bc bd 2.6. f(x) = px 2 + qx + r p, q, r r 0 X = 0, 1, 1 h(f(0)) = 2h(0) = 0 1.5(2) f(0) = r 0, 1, 1 f(1) = p + q + r f( 1) = p q + r 0, 1, 1 (p, q, r) = (p, q, r) = (1, 0, 0), ( 1, 0, 0) f(0) = 0, f(1) = 1, f( 1) = 0 (p, q, r) = ( 1 2, 1 2, 0), f(x) = 1 2 X X 27 f(2) = 3 h(f(2)) = log(3) 2 log(2) = 2h(2) f(x) (p, q, r) = ( 1 2, 1 2, 0) 3.3. f n (X) = X 2n n f- ( ) ( ) 2π 2π x = cos 2 n + i sin 1 2 n x f x f 2 1(x) = x f 2 1(X) X = X 4 2X 2 X = X(X + 1)(X 2 X 1) x = 0 1 y y y 2 1 = 0 y = ±1 y 2 1 = 1 y = 0 y y 2 1 = 1. y f 1 -y = 0, 1, x f- n n 3 x 0,..., x n 1 x 0 = x, x i+1 = f i (x i ), x 0,..., x n 1

40 B 40 2 g(x, Y ) (B.1) f(y ) f(x) Y X = g(x, Y ) 4.7 g(x, Y ) = X + Y. i = 0,..., n 1 x i+2 x i+1 x i+1 x i = g(x i, x i+1 ) n 1 i=0 g(x i, x i+1 ) = 1 g(x i, x i+1 ) i g(x i, x i+1 ) 1 1 i g(x i, x i+1 ) = 1 (B.1) x i+2 = x i i g(x i, x i+1 ) = 1 (B.1) x 0 = = x n 1 n ( 1, 0) C : X 2 + Y 2 = 1 (x, y) ( 1, 0) ( 1, 0) (x, y) l t t l : Y = t(x + 1) (x, y) C l ( 1, 0) C l x = 2t 1 + t 2, y = 1 t2 1 + t (1) l Y = 3a2 2b X + 3a3 + 2b 2. 2b (2) l E Y b 2 = a 3 2 R X a4 +16a 2 4(a 3 2) = a4 +16a 2 4b 2 4(a 3 2)X 3 9a 4 X 2 + (6a a 2 )X (a a 3 ) = 0 (X a) 2 ( 4(a 3 2)X (a a 2 ) ) = 0. R Y a 2 2b a a 2 4b 2 + 3a3 + 2b 2 = a6 40a b 4b 3 (1) f(x) = 0 n c d = n(c 0 n d 1 + c 1 n d c d 1 )

41 41 n c d c d. 1 m c d m c d m. c d m 1,..., m k m i f(m 1 ),..., f(m k ) 0 f(m i ) = 0 i (i = 1,..., k) f(x) = 0 i f(x) = 0 (2) Kroneker β 0,..., β l l g(x) g(0) = β 0,..., g(l) = β l (B.2) g(x) = l i=0 ( 1) l i β i i!(l i)! X (X i + 1)(X i 1) (X l) g(x) l g(0) = β 0,..., g(l) = β l g 1 (X), g 2 (X) g 1 (X) g 2 (X) l (l + 1) 0, 1,..., l 0 g 1 (X) g 2 (X) 0 g(x) f(x) = g(x)h(x) g(x), h(x) g(x) l 1 l d i = 0, 1,..., l f(i) = g(i)h(i) f(i), g(i), h(i) g(i) f(i) 1 l d l f(i) i = 0,..., l m i1,..., m iki, m ij l g(x) g(i) = m 0j0,..., g(l) = m ljl 1 j 0 k 0,..., 1 j l k l (B.2) g(x) f(x) l 1 l d m iji 0 i l, 1 j i k i g(x) g(x) ±1, ±f(x), g(x) f(x) f(x) g(x) f(x) [1] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag [2] J. Milnor, Dynamics in one complex variable. Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig [3] J. H. Silverman The arithemtic of dynamical systems, Graduate Texts in Mathematics 241, Springer- Verlag [1] [2] [3]

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,