S-integral points on hyperelliptic curves Homero Renato Gallegos ...

Theorem 3.7.1. Let S be a finite set of rational primes of cardinality s. Let P be thelargest prime in S, with the convention that P = 1 if S is empty. Let α be an algebraic integerof degree at least 3, and let κ be an integer belonging to K = Q(α). Let α 1 , α 2 , α 3be different conjugates of α and let κ 1 , κ 2 , κ 3 be the corresponding conjugates of κ. LetK 1 = Q(α 1 , α 2 , √ κ 1 κ 2 ), K 2 = Q(α 1 , α 3 , √ κ 1 κ 3 ), K 3 = Q(α 2 , α 3 , √ κ 2 κ 3 ),andL = Q(α 1 , α 2 , α 3 , √ κ 1 κ 2 , √ κ 1 κ 3 ).Let d 1 , d 2 , d 3 and r 1 , r 2 , r 3 be the degrees and the unit ranks of K 1 , K 2 , K 3 respectively.Let d be the degree of L. Let R be an upper bound for the regulators of K 1 , K 2 , K 3and R S an upper bound for the respective S Ki -regulators of K 1 , K 2 , K 3 . Let s i be thenumber of places in S Ki . Let h be an upper bound for the class numbers of the K i . Letc ∗ j = maxi=1,2 c j(s i , d i ), j = 1, . . . , 5,c ∗ 6 = max c 6(r i , d i ),i=1,2,3 ∣N = max ∣Norm Q(αi ,α1≤i,j≤3j )/Q(κ i (α i − α j )) ∣ 2 ,⎧⎛ ⎞⎫⎨H ∗ log N= max+ h ⎝ ∑ ⎬log p⎠ + c ∗⎩min i=1,2,3 d6R + h(κ), 1, π/di ⎭ .p∈Sc ∗ 10 = 2H ∗ + 2H ∗ d(s + 1)(1 + 2(c ∗ 4) 2 c 7 (s 1 + s 2 − 1, d)R 2 S×log( √ 2e max{(s 1 + s 2 − 2)π/ √ 2, c ∗ 2R S }),c ∗ 11 = 4d(s + 1)H ∗ (c ∗ 4) 2 c 7 (s 1 + s 2 − 1, d)RS2( )max{cc ∗ 12 = 2H ∗ + 2H ∗ (d(s + 1)) + c ∗ ∗11 log5 , 1}2 √ ,2dH ∗c ∗ 13 = log 2 + 2H ∗ + 4(s 1 + s 2 − 2)H ∗ (c ∗ 1) 2 c ∗ 2c 9 (s 1 + s 2 − 1, d)R 3 S38