Download PDF by B.M.M. de Weger: Algorithms for Diophantine Equations

By B.M.M. de Weger

ISBN-10: 9061963753

ISBN-13: 9789061963752

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B = | o | g | [gWCWy ] ... , b are a basis of the lattice. Then G 1 n n n-1 is a sublattice of Z of determinant g W[gWCWy ] , which is of size C . , gWx , ~L )T , S x Wb = i i 9 1 n-1 0 i=1 are integers, and n S x W[gWCWy ] . i i i=1 53 ~ L Clearly, both is close to gWCWL . The length of the vector x now measures X and |L| , which are exactly the two numbers we want to balance 0 with each other. Heuristics (cf. 3) tell us that in a generic case -n we expect |L| = X . We now can prove easily the following useful lemma.

2) it follows that i-1 d i-1 S -----------------------------------Wl Wc . , i-1 j d j S -----------------------------------Wl Wc . d Wd i,k k k=1 k-1 k c (j) = d Wb i j i Then c (0) = b , and c (i-1) = c . The i i i i computed in (A) at the j th step, since c (j) i is exactly the vector d Wc (j-1) - l Wc j i i,j j ---------------------------------------------------------------------------------------------------d j-1 = d Wb j i j-1 d d j j S -----------------------------------Wl Wc - -----------------------------------Wl Wc = c (j) .

For a good survey of this field, see Brentjes [1981]. However, the available algorithms in this field seem not to have the desired efficiency and generality. Fortunately, since 1981 there is a useful alternative, which in a sense is also a generalization of the one-dimensional continued fraction algorithm. In 1981, L. Lovssz invented an algorithm, that has since then become known as 3 the L -algorithm. It has been published in Lenstra, Lenstra and Lovssz [1982], Fig. 1, p. 521. Throughout this and the next section we refer to this paper as "LLL".

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

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