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Siegel's lemma is sharp for almost all linear systems
Author(s) -
Baker Roger,
Masser David
Publication year - 2019
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12281
Subject(s) - mathematics , upper and lower bounds , combinatorics , lemma (botany) , zero (linguistics) , mathematical analysis , ecology , linguistics , philosophy , poaceae , biology
The well‐known Siegel Lemma gives an upper bound c U m / ( 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 boundU m / ( n − m ) / B is relatively rare for large B ⩾ 1 ; for example, there are θ = θ ( m , n ) > 0 andc ′ = c ′ ( m , n )such that this happens for at mostc ′ U m n / B θout of the roughly( 2 U ) m npossible such systems.