Siegel's lemma is sharp for almost all linear systems

The well‐known Siegel Lemma gives an upper bound cUm/(n−m) for the size of the smallest non‐zero integral solution of a linear system of m⩾1 equations in n>m unknowns whose coefficients are integers of absolute value at most U⩾1; here c=c(m,n)⩾1. In this paper, we show that a better upper bound U...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2019-10, Vol.51 (5), p.853-867
Main Authors: Baker, Roger, Masser, David
Format: Article
Language:English
Subjects:
11J13 (primary)
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Summary:The well‐known Siegel Lemma gives an upper bound cUm/(n−m) for the size of the smallest non‐zero integral solution of a linear system of m⩾1 equations in n>m unknowns whose coefficients are integers of absolute value at most U⩾1; here c=c(m,n)⩾1. In this paper, we show that a better upper bound Um/(n−m)/B is relatively rare for large B⩾1; for example, there are θ=θ(m,n)>0 and c′=c′(m,n) such that this happens for at most c′Umn/Bθ out of the roughly (2U)mn possible such systems.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms.12281