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This is now an important 44 Logarithmic forms instrument in transcendence theory and its discovery and development has been a major achievement; see the discussion beginning in Chapter 4. 1. 1 is consequently stronger. Recently some work of Matveev [174] has appeared which gives an improved form for the expression for n of the shape cn for an absolute constant c. Matveev’s articles contain a number of new elements and they constitute an important advance; for a discussion in the simplest case see the article by Nesterenko in [3, pp.

D is an integral basis for K and a1 , . . , ad are rational integers with absolute values A. The latter estimate follows at once from the equation and its ﬁeld conjugates which enable one to express each aj as a linear combination of the conjugates of α. This indeed is the technique used in establishing a generalised version of Siegel’s lemma to number ﬁelds [25, Ch. 1. Then √ we would need only the √ condition N > M and we could take L = 2hk instead of L = 2dhk . In any event, our √ k whence N = (L + 1)2 M and choice of L ensures that L LR k.

Yr can be expressed as linear combinations of log y(1) , . . , log y(r) with coefﬁcients given by minors of order (r − 1) of R. This gives max log y( j) . Y Y or Let the maximum be given by j = l; then either log y(l) (l) log y −Y . In the ﬁrst case we have the desired assertion. In the second case we recall that y is a unit and thus d log y( j) = 0. j=1 Hence we have log y Y for some conjugate y of y as asserted. 1 for the general system of S-units US we denote by p1 , . . , ps the prime ideals corresponding to the ﬁnite places of S.