Abstract
A vector m=(m 1,...,m n ) ∈ Z n \ {0} is called an integer relation for the real numbers α 1,...,α n , if Σα i m i =0 holds. We present an algorithm that when given algebraic numbers α 1,...,α n and a parameter ɛ either finds an integer relation for α 1,...,α n or proves that no relation of euclidean length shorter than 1/ɛ exists. Each algebraic number is assumed to be given by its minimal polynomial and by a rational approximation precise enough to separate it from its conjugates.
Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs
bit operations. The straightforward algorithm that works with a primitive element of the field extension Q(α 1,...,α n ) of Q would take poly (n, log maxi height(α i ), Π i=1n degree (α i )) bit operations.
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© 1989 Springer-Verlag Berlin Heidelberg
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Just, B. (1989). Integer relations among algebraic numbers. In: Kreczmar, A., Mirkowska, G. (eds) Mathematical Foundations of Computer Science 1989. MFCS 1989. Lecture Notes in Computer Science, vol 379. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51486-4_78
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DOI: https://doi.org/10.1007/3-540-51486-4_78
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