F = 0 F α, β F = t 2 + at + b (t α)(t β) = t 2 (α + β)t + αβ G : α + β = a, αβ = b F = 0 F (t) = 0 t α, β G t F = 0 α, β G. α β a b α β α β a b (α β)

19 7 12 1 t F := t 2 + at + b D := a 2 4b F = 0 a, b 1.1 F = 0 α, β α β a, b /stlasadisc.tex, cusp.tex, toileta.eps, toiletb.eps, fromatob.tex 1

F = 0 F α, β F = t 2 + at + b (t α)(t β) = t 2 (α + β)t + αβ G : α + β = a, αβ = b F = 0 F (t) = 0 t α, β G t F = 0 α, β G. α β a b α β α β a b (α β) 2 (α β) 2 = (α + β) 2 4αβ = a 2 4b = D. 1.2 t t + c c s = F (t) (t, s)- s t = 0 t 2 + at + b = ( t + a ) 2 a 2 4b 2 4 = ( t a ) 2 D 2 4. 1.3 t = c F F (c) s = F (t) (c, F (c)) c F = 0 (c, 0) t- F (c) = 0, F (c) = 0 F = 0 c R : F (c) = 0, F (c) = 0 2

R : c 2 + ac + b = 0, 2c + a = 0 c = a/2 0 = ( a ) 2 ( + a a ) + b = D 2 2 4 D F (t) = 0, F (t) = 0 t F D ±D 2 t F := t 3 + pt 2 + qt + r F = 0 p, q, r 2.1 F = 0 α, β, γ (α β)(β γ)(γ α) p, q, r F = 0 F α, β, γ (t α)(t β)(t γ) = t 3 (α + β + γ)t 2 + (αβ + βγ + γα)t αβγ F = t 3 + pt 2 + qt + r G : α + β + γ = p, αβ + βγ + γα = q, αβγ = r 3

F = 0 F (t) = 0 t α, β, γ G t F = 0 α, β, γ G. α β p, q r (α β)(β γ)(γ α) p, q r D = (α β) 2 (β γ) 2 (γ α) 2 ( α, β, γ p, q, r 2.2 t t + c c t 3 + pt 2 + qt + r = ( t + p ) ) 3 + (q p2 t p3 3 3 27 + r. t F F = t 3 + qt + r p = 0 2.3 F (t) = 0, F (t) = 0 t t 3 + qt + r = 0, 3t 2 + q = 0 4

t 2 = q/3 0 = q 3 t + qt + r = 2 3r qt + r t = 3 2q. 4q 3 + 27r 2 = 0 t D = (α β) 2 (β γ) 2 (γ α) 2 p, q, r D 4q 3 + 27r 2 q, r q = αβ + βγ + γα, r = αβγ α + β + γ = 0 D = (4q 3 + 27r 2 ) D > 0 D < 0 2.4 (q, r)- D = 4q 3 + 27r 2 = 0 parameter (t α) 2 (t β), 2α + β = 0 F = t 3 + qt + r q = 3α 2, r = 2α 3 α 1 α α ᾱ z = x+iy z = x iy z, w z + w = z + w, zw = z w 5

1 + 2 r 1 + (1 + 1) 1 + 1 + 1 3 q 2 + 1 1: qr D = 0 1 + 1 + 1 1 + (1 + 1) 2 + 1 1 + 2 3 D = 0 1 + 1 + 1 1 + (1 + 1) D = 0 3 1 + 2, 2 + 1 1 + 1 + 1 1 + (1 + 1) 2 \ / 2 + 1 1 + 2 1 \ / 3 0 (q, r)- 2 + 1, 1 + 2 1 + 1 + 1 1 + (1 + 1) 2 1 2 + 1, 1 + 2 6

1 + 2 1 + 1 + 1 3 1 + (1 + 1) 2 + 1 2: qr D = 0 2.5 F = t 3 + qt + r q, r α, β, γ = α β t = α + β, q = αβ + βγ + γα = αβ (α + β) 2, r = αβγ = αβ(α + β) s = αβ q = s t 2, r = st 4q 3 + 27r 2 = (s + 2t 2 ) 2 (4s t 2 ) (s, t) (q, r) = (s t 2, st) (s, t) 3 ; 3 t F F = 0 t t + c c F F = t 4 + xt 2 + yt + z 7

r t e d c b a s c e b a q d 3.1 3: (s, t) (q, r) = (s t 2, st) F = 0 α, β, γ, δ = (α β)(α γ)(α δ) (β γ)(β δ) (γ δ) 2 D α + β + γ + δ = 0, αβ + αγ + αδ + βγ + βδ + γδ = x, βγδ + αγδ + αβδ + αβγ = y, αβγδ = z D x, y, z F (t) = 0, F (t) = 0 t t 4 + xt 2 + yt + z = 0, 4t 3 + 2xt + y = 0 t t t t 3 t t 4 = t t 3 t t 2 8

t t 3 = t t 2 t t t 2 t x, y, z t 256z 3 128x 2 z 2 + (144y 2 x + 16x 4 )z 4y 2 x 3 27y 4 D z z/4 F = t 4 + xt 2 + yt + z 4, D := 4z 3 8x 2 z 2 + (36y 2 x + 4x 4 )z (27y 2 + 4x 3 )y 2. 3.2 xyz- D = 0 S t 4 + xt 2 + yt + z/4 = 0 1 + 1 + 1 + 1 1 + 1 + (1 + 1) (1 + 1) + (1 + 1) 3 1 + 1 + 2 1 + 2 + 1 2 + 1 + 1 2 + (1 + 1) 2 1 + 3 3 + 1 2 + 2 2(1 + 1) 1 4 0 t 4 (x, y, z) = (0, 0, 0) 1 + 1 + 1 + 1 1 + 1 + (1 + 1) 9

(1 + 1) + (1 + 1) 1 + 1 + 2 1 + 2 + 1 2 + 1 + 1 2 + (1 + 1) 1 + 3 3 + 1 2 + 2 2(1 + 1) 4. (x, y, z)- S 2(1 + 1) (1 + 1) + (1 + 1) S : 1 + 1 + 2, 1 + 2 + 1, 2 + 2 2 + (1 + 1) S 1 + 1 + 1 + 1 (1 + 1) + (1 + 1) 1 + 1 + (1 + 1) 4 3.3 S 3.3.1 2 + 2 γ = α, δ = β (t α) 2 (t β) 2, 2α + 2β = 0 t 4 2α 2 t 2 +α 4 F = t 4 +xt 2 +yt+z/4 = 0 P : (x, y, z) = (x, 0, x 2 ), x < 0 10

1 + 2 + 1 2 + 1 + 1 1 + 1 + 2 2 + (1 + 1) 4: xyz S : D = 0 3.3.2 1 + 3, 3 + 1 γ = δ = α (t α) 3 (t β), 3α + β = 0 F C : (x, y, z) = ( 6α 2, 8α 3, 12α 4 ) xy- z 3.3.3 2(1 + 1) α = β = iη, γ = δ = iη (t iη) 2 (t + iη) 2 = (t 2 + η 2 ) 2 = t 4 + 2 η 2 t 2 + 4 η 4 4 F P + : (x, y, z) = (x, 0, x 2 ), x > 0 11

3.4 S : D = 0 D = 0 S δ = γ (t α)(t β)(t γ) 2, α + β + 2γ = 0 F = t 4 + xt 2 + yt + z/4 x = 3 4 α2 1 2 αβ 3 4 β2, y = 1 4 (α + β)( β + α)2, z = (α + β) 2 αβ α, β α, β α < β < γ 1 + 1 + 2 α, β α < γ < β 1 + 2 + 1 α, β γ < α < β 2 + 1 + 1 α, β 2 + (1 + 1) x, y, z α, β α β α + β = a, αβ = b a, b x = 3 4 a2 + b, y = 1 4 a(a2 4b), z = 4a 2 b ab- 5 α = β a 2 4b = 0, α = γ 3a 2 + 4b = 0. a, b 6 : x α = 3 2 a, x β = 1, y α = 1 4 (3a2 4b), y β = a, z α = 2ab, z β = a2. 12

b 2 + 2 a 2 4b = 0 2 + (1 + 1) a 1 + 1 + 2 2 + 1 + 1 1 + 2 + 1 3a 2 + 4b = 0 1 + 3 3 + 1 5: S (a, b) 6: ( 1 + 3 1 + 1 + 2 1 + 2 + 1 3 + 1 2 + 1 + 1 1 + 2 + 1 2 + 2 1 + 1 + 2 2 + 1 + 1 1 + 1 + 2 2 + (1 + 1) 2 + 1 + 1 2 + (1 + 1) 7 swallowtail P P P C 13

P C 7: 4 14