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Some comments about indefinite LLL. (English) Zbl 1353.11059
Chan, Wai Kiu (ed.) et al., Diophantine methods, lattices, and arithmetic theory of quadratic forms. Proceedings of the international workshop, Banff International Research Station (BIRS), Alberta, Canada, November 13–18, 2011. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8318-1/pbk; 978-0-8218-9503-0/ebook). Contemporary Mathematics 587, 233-243 (2013).
The classical LLL algorithm [A. K. Lenstra et al., Math. Ann. 261, 515–534 (1982; Zbl 0488.12001)] inputs a symmetric nonsingular, definite Gram matrix \(G\) of dimension \(n\) and from that calculates a Gram matrix corresponding to the reduced basis. For non-singular matrices there are several extensions of the LLL algorithm, for example the modified LLL algorithm of M. Pohst [J. Symb. Comput. 4, 123–127 (1987; Zbl 0629.10001)]. D. Simon [Math. Comput. 74, No. 251, 1531–1543 (2005; Zbl 1078.11072); “Quadratic equation in dimension 4,5 and more”, Preprint (2005)] considers the LLL algorithm with indefinite matrices.
In the present paper the author exposes and comments the work of D. Simon.
For the entire collection see [Zbl 1260.11002].
11D09 Quadratic and bilinear Diophantine equations
11Y50 Computer solution of Diophantine equations