Read e-book online Algorithms for Diophantine Equations PDF

By B.M.M. de Weger

ISBN-10: 9061963753

ISBN-13: 9789061963752

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N . , n is an arbitrary but fixed determination of j + the logarithm of a . Let V = max(V ,1) for j = n, n-1 , and put j j j 29 n S bjWlog aj . L = Put B = j=1 max |b | . WVnWlog(eWDWV+n-1)W W(9 log B + log(eWDWV+n) )0 )0 , ( 8Wn + 51, 10Wn + 33, 9Wn + 39 ) . ,ra ):Q] = 2 , then we can take e(n) = 9Wn + 26 and 1 n 2Wn n+4 replace the factor n in the above bound for |L| by n . Waldschmidt’s main theorem does not give the constant e(n) as detailed as we do, but he does so in his proof, cf. p. 283.

0 We may assume that x h is the largest solution of x = a + bW(log x) By 1/h 1/h < z1/h + z 1 2 1/h < a1/h + cWlog(x1/h) , (z +z ) 1 2 x where . 1/h c = hWb . Define we infer y 1/h by x = (1+y)WcWlog c . From log c < log(cWlog c) it follows that h h ( ( h h))h c W(log c) < bW log c W(log c) , 9 which implies 9 00 h h x > c W(log c) . Hence 1/h (1+y)WcWlog c = x y > 0 . Now, < a1/h + cWlog(1+y) + cWlog c + cWloglog c 1/h < a + cWy + cWlog c + cWloglog c . Hence 1/h yWcW(log c - 1) < a If 2 c > e + cWloglog c .

K a _ 0 (mod p ) . The p-adic norm is defined . 1 we have seen how to define ord ord on algebraic p extensions of Q . For any a e W with ord (a) > 1/(p-1) we can define p p the p-adic logarithm log (1+a) by the Taylor series p p and 2 3 log (1+a) = a - a /2 + a /3 - ... p This logarithmic function has the well known properties of a logarithm, such log (x Wx ) = log (x ) + log (x ) for all x , x for which it is p 1 2 p 1 p 2 1 2 defined. Further, log (x) = 0 if and only if x is a root of unity.

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

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