## Computation of highly regular nearby points

`We call a vector x/spl isin/R/sup n/ highly regular if it satisfies =0 for some short, non-zero integer vector m where <...> is the inner product. We present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//s`
`We call a vector x/spl isin/R/sup n/ highly regular if it satisfies =0 for some short, non-zero integer vector m where <...> is the inner product. We present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/N finds a highly regular nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no short relation m~ of length less than /spl alpha//2 exists for points x~ within half the x'-distance from x. The integer relation m for x' is for random x up to an average factor 2/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x.`

Author: Carsten Rössner, Claus Peter Schnorr urn:nbn:de:hebis:30-12331 Article English 2005/07/13 1995 Univ.-Bibliothek Frankfurt am Main 2005/07/13 computational complexity ; computational geometry ; highly regular nearby points ; inner product ; integer vector ; short integer relation 3rd Israel Symposium on the Theory of Computing and Systems, 1995 , http://www.mi.informatik.uni-frankfurt.de/research/papers.html 224772023 Mathematik Informatik 510 Mathematik Universitätspublikationen Veröffentlichungsvertrag für Publikationen

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