By B.M.M. de Weger
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Additional resources for Algorithms for Diophantine Equations
In our applications we often have a lattice A such that the associated matrix, ( 1 | . o | . A = | o 1 | 7 Q1 ... Qn-1 Qn where the Q space split G , of which a basis is given say, has the special form ) | | | , | 8 are large integers, that may have several hundreds of decimal i digits. We can compute a reduced basis of this lattice directly, using the 3 matrix A itself as input for the L -algorithm. But it may save time and to up the computation into several steps with increasing accuracy, as follows.
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) .
In this section we mention the facts about p-adic numbers and functions that we use. For details we refer to Bachman  and Koblitz , . 27 We assume that the reader is familiar with the field of p-adic numbers and the p-adic valuation . 1. e. the field to which all p-adic theory is applied. , p-1 } , p i with u $ 0 . The number 0 can be represented in this way by taking k = 0 k and all digits equal to 0 , and ord (0) = 8 by definition. If ord (a) > 0 p p then a is called a p-adic integer.
Algorithms for Diophantine Equations by B.M.M. de Weger