Siegel’s lemma
E871398
Siegel’s lemma is a result in number theory that guarantees the existence of small-height integer solutions to systems of linear equations with integer coefficients.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Siegel’s lemma canonical | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf | lemma in number theory ⓘ |
| appearsIn | Carl Ludwig Siegel’s work on Diophantine equations ⓘ |
| appliesTo | linear forms over number fields (in generalized versions) ⓘ |
| assumes |
homogeneous linear equations
ⓘ
integer coefficient matrix ⓘ |
| category | results on small solutions of linear systems ⓘ |
| concerns |
integer solutions of linear equations
ⓘ
systems of linear equations with integer coefficients ⓘ |
| concludes | existence of a nonzero integer vector in the kernel ⓘ |
| context |
Diophantine geometry
NERFINISHED
ⓘ
arithmetic geometry ⓘ |
| field | number theory ⓘ |
| formalizes | existence of short integer relations among vectors ⓘ |
| generalizationOf | earlier results in geometry of numbers on small solutions ⓘ |
| guaranteesExistenceOf |
nontrivial integer solutions of homogeneous linear systems
ⓘ
small-height integer solutions ⓘ |
| hasVariant |
Bombieri–Vaaler version of Siegel’s lemma
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ⓘ
absolute Siegel’s lemma NERFINISHED ⓘ p-adic Siegel’s lemma NERFINISHED ⓘ |
| influenced | development of effective Diophantine methods ⓘ |
| involvesConcept |
bounds on solutions in terms of coefficients
ⓘ
height of an integer vector ⓘ |
| isToolFor |
bounding heights in projective space
ⓘ
constructing small-height bases of lattices ⓘ |
| namedAfter | Carl Ludwig Siegel NERFINISHED ⓘ |
| oftenExtendedTo | number fields ⓘ |
| oftenFormulatedOver | rational numbers ⓘ |
| provides | upper bounds on the size of integer solutions ⓘ |
| relatedTo |
Minkowski’s theorem
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ⓘ
Subspace theorem NERFINISHED ⓘ Thue–Siegel–Roth theorem NERFINISHED ⓘ |
| standardReference |
Bombieri and Gubler’s Heights in Diophantine Geometry
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ⓘ
Cassels’ An Introduction to Diophantine Approximation NERFINISHED ⓘ Serge Lang’s books on Diophantine approximation NERFINISHED ⓘ |
| status | classical result in number theory ⓘ |
| subfield |
Diophantine approximation
ⓘ
geometry of numbers ⓘ |
| typicalBoundDependsOn |
maximal absolute value of coefficients
GENERATED
ⓘ
number of equations GENERATED ⓘ number of variables GENERATED ⓘ |
| typicalSetting | more variables than equations ⓘ |
| usedBy |
theory of linear recurrences and relations
ⓘ
transcendence theory ⓘ |
| usedIn |
bounds for solutions of Diophantine equations
ⓘ
proofs in Diophantine approximation ⓘ results on linear forms in logarithms ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.