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By Rudenskaya O.G.

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Fortunately it follows from a result of Mignotte (1974) that (g) < 2n{/) , so this is easily achieved. We conclude with an application of these considerations. The following result was proved by A. K. Lenstra, H. W. Lenstra and L. Lovasz (1982) in a different way. The algorithm given here was sketched in that paper and elaborated on by R. Kannan, A. K. Lenstra and L. Lovasz (1984). 8) Corollary. A polynomial with rational coefficients can be factored into irreducible polynomials in polynomial time.

Why not the Dedekind cut model? One could do the latter; indeed then the oracle would accept rational numbers r as its input, and answer "my real number is larger/ not larger than r ". The two models are not equivalent (cf. Ko (1983)); the "Cauchy sequence oracle" is weaker, and yet sufficient for at least those results which follow. We shall only address one question concerning real number boxes: how do special classes of real numbers fit in? There is no problem with integers. First, if a is any integer then we may design a box which answers "a" to any query; as output-guarantee we can write "{r) < (a) • (e)" on it.

Hor using weak reducedness it is easy to see that this is the same as saying that b{(i) is not longer than any vector in the lattice generated by bi(i] and 6j+i(i) . This observation suggests the idea of fixing any integer k > 1 and saying that the lattice L is k-reduced if it is weakly reduced and, for all 1 < i < n , (if i + k > n , then we disregard the undefined vectors among the generators on the right hand side). 13) Theorem. Let ( 6 1 , . . , 6 n ) be a k-reduced basis of the lattice L .

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