Lecture Notes in Computer Science Volume 210, 1986, pp 105-118

Polynomial time algorithms for finding integer relations among real numbers

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We present algorithms, which when given a real vector x∈ℝn and a parameter k∈ℕ as input either find an integer relation m∈ℤn, m≠0 with xTm=0 or prove there is no such integer relation with ‖m‖≦2k. One such algorithm halts after at most O(n3(k+n)) arithmetic operations using real numbers. It finds an integer relation that is no more than \(2^{\frac{{n - 2}}{2}}\) times longer than the length of the shortest relation for x. Given a rational input x∈ℚn this algorithm halts in polynomially many bit operations. The basic algorithm of this kind is due to Ferguson and Forcade (1979) and is closely related to the Lovàsz (1982) lattice basis reduction algorithm.