Algebraic Geometry, la Rabida, Spain 1981: Proceedings by J. M. Aroca, R. Buchweitz, M. Giusti, M. Merle

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By J. M. Aroca, R. Buchweitz, M. Giusti, M. Merle

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Moscow 1952 (Russian). , New York 1960. [4] W. J. LeVeque, On the equation y m = f (x). Acta Arith. 9 (1964), 209–219. [5] V. G. Sprindzhuk, An effective analysis of the Thue and Thue–Mahler equations. In: Current Problems of Analytic Number Theory (Minsk 1972), Nauka i Tekhnika, Minsk 1974, 199–222 (Russian). [6] R. Tijdeman, Applications of the Gel fond–Baker method to rational number theory. In: Topics in Number Theory (Debrecen 1974), Colloq. Math. Soc. János Bolyai 13, North-Holland, Amsterdam 1976, 399–416.

If X = Y we have (ax − aα1 )h = (ax − aα2 )h and it follows, from α1 = α2 , that ax − aα1 = e2πig/ h (ax − aα2 ), 0 < g < h, and |x| |α1 | + |α2 | . 2 sin(π/ h) A8. On the equation y m = P (x) 45 Since |y| > 1, equation (1) gives m < c1 , where c1 as the subsequent constants c2 , c3 , . . depends only on P and is effectively computable. log XY −1 If X = Y we have either |X| = |Y | or |X| = |Y | and = 0. In the former 2π i case we infer by (8) from Baker’s theorem [2] that log |XY −1 | > H |γ1 /γ2 |2 −c2 log s , in the latter case similarly log XY −1 > H γ1 /γ2 2πi −c3 log s , where in case H ( ) = 1, it should be replaced by 2.

We will show that d = ±1. Set B = d · A−1 . Then B = (bij ) is an integral matrix of determinant d 7 , and g(X · B) = f (dX), which can be rewritten to give norm(X1 λ1 + . . + X8 λ8 ) = d 8 · f (X) (2) where λi = j bij θj . If A is the integral ideal generated by the λ’s, then (2) implies (3) norm A d 8. Consider the lattice Λ = Zλi contained in A. Then B is the matrix of the transition map from the lattice O = Zθi of all integers in K to the lattice Λ, so the index of Λ in O is [O : Λ] = |det B| = |d|7 .

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