By B.M.M. de Weger
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Extra resources for Algorithms for Diophantine Equations
4_(Waldschmidt). , b e Z ( n > 2 ) . Let 1 n 1 n positive real numbers satisfying 1/D < V < ... < V and 1 n V j where > max (9 h(aj), |log aj|/D log a ) 0 for [K:Q] = D . , 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 .
_(Davenport). y with q > X 0 . We have the following result. 10) we denote the distance to the nearest integer). 2) satisfy 1 ( 2 ) . X < -----Wlog q Wc/|y |WX d 9 2 00 Proof. 10) we infer 2WX /q < NqW(j-x Wy+x )+x W(qWy-p)N < qW|L/y | + |x |/q . 11). 10) is not true for the first convergent with denominator one should try some further convergents. 11) yields a reduced upper bound for X of size log X , 0 0 as desired. 10) (a situation which is very unlikely to occur, as experiments show), then not all is lost, since then only very few exceptional possible solutions have to be checked.
Put U = I , B = A . For some i i 0 1 1 3 j > 1 let B and U be known matrices. Then we apply the L -algorithm j j-1 -1 to B = B , U = U , and U . We thus find matrices C , U , and j j-1 j j -1 Uj such that Notice that C6j Ak = A , since BjWU-1 WU j-1 j = . Now put Bj+1 By induction EWCj = Bj + Cj , Bj+1WU-1 j = DjWUj and . , k . Note that , -1 so the BjWUj-1 satisfy the same recursive relation as the -1 B1WU0 = A1 , we have BjWU-1 = A for all j . Hence j-1 j Cj BjWU-1 WU6 j-1 j = = Ck and it follows that AjWUj and Aj .