Computation of Highly Regular Nearby Points
نویسندگان
چکیده
We call a vector x 2 IR n highly regular if it satisses < x ; m > = 0 for some short, non{zero integer vector m where < : ; : > is the inner product. We present an algorithm which given x 2 IR n and 2 IN nds a highly regular nearby point x 0 and a short integer relation m for x 0 : The nearby point x 0 is 'good' in the sense that no short relation m of length less than =2 exists for points x within half the x 0 {distance from x: The integer relation m for x 0 is for random x up to an average factor 2 n=2 a shortest integer relation for x 0 : Our algorithm uses, for arbitrary real input x; at most O(n 4 (n + log)) many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most
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