By B.M.M. de Weger

ISBN-10: 9061963753

ISBN-13: 9789061963752

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Extra resources for Algorithms for Diophantine Equations

Example text

There be an element of degree d , and > 0 be the leading coefficient of its minimal polynomial over 0 We define the (logarithmic) height h(a) by 1 Wlog(9a0D/dWpmax(1,|s(a)|))0 , D s h(a) = ----- where the product is taken over all embeddings does not depend on the field s . Note that this definition K . , a , then the above definition applied for 1 d Then of s Dirichlet’s K a e Q , then with a = p/q h(a) = log max(|p|,|q|) , and if r = s + t - 1 real and Unit 2Wt Theorem independent units is given by a are yields ----- In particular, if Let there be K = Q(a) a d 1 ( ) W log a W p max(1,|a |) .

The result follows. p If The next lemmas make explicit that x and log(1+x) are near if |x| is small in the real and complex case, respectively. 2. a e R . If Let a < 1 |x| < a and then |log(1+x)| < -log(1-a) W|x| , a --------------------------------------------- and a |x| < Proof. 1-e Note that log(1+x)/x is a strictly positive and strictly decreasing |x| < 1 . Hence it is for function for at W|ex-1| . -a ------------------------- |x| < a x = -a . 3. 0 < a < p . If Let |x| < a always less than its value x x/(e -1) .

2). Therefore, + c Wm < m , 2 j so that we find a new upper bound for m , that is of the size of m , which j is about log N / log p . We repeat this procedure for all the m , in 0 j order to obtain a reduced upper bound for H . If this is not yet sufficient p to derive at once a reduced upper bound for H , then we can do so by applying a reduction step for real linear forms, where we may take advantage of the fact that for some of the variables a much better upper bound has just been found (cf.

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Algorithms for Diophantine Equations by B.M.M. de Weger

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