By B.M.M. de Weger
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Extra resources for Algorithms for Diophantine Equations
5). 1(i) directly, or as follows. 2. 7). 1) , then 1 1 ( ) 1 X < -----Wlog cW(A+2)/|y | + -----Wlog X . d 9 2 0 d Remark. 1 is sharp for large X Proof. b follows. We can apply Lemma only. 5) yield (a 2 -1 > q W|y |/|L| > q W|y |Wc Wexp(dWX) . 1(i). In practice it does not often occur that A p is large. Therefore this lemma is useful indeed. Summarizing, this case comes down to computing the continued fraction of a real number to a certain precision, and establishing that it has no extremely large partial quotients.
ZWa is called an order of K if it is a subring of the 1 D ’maximal order’ O . K any algebraic integer can be written as a product of irreducible elements. Here an irreducible element (prime element) is an element that has no integral divisors but its own associates. However, this decomposition into primes need not be unique. Ideals can also be decomposed into prime ideals, and this decomposition is unique. A principal ideal is an ideal generated by a single element a . Two fractional ideals are called equivalent if their quotient is principal.
LW(91+(k-1)/n)0 instead of lWk Finding all short lattice points: the Fincke and Pohst algorithm. Sometimes it is not sufficient to have only a lower bound for l(G,y) . It may be useful to know exactly all vectors |x| < C or |x-y| < C for a given constant x e G l(G) or such that C . There exists an efficient algorithm for finding all the solutions to these problems. This algorithm was devised by Fincke and Pohst , cf. 12). We give a description of this algorithm below. B The input of the algorithm is a matrix lattice points G , and a constant x e G with and x i C > 0 .