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 (4)
| Label | Occurrences |
|---|---|
| Bombieri–Vaaler version of Siegel’s lemma | 1 |
| Siegel’s lemma canonical | 1 |
| absolute Siegel’s lemma | 1 |
| p-adic Siegel’s lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10543861 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Siegel’s lemma Context triple: [Carl Ludwig Siegel, notableWork, Siegel’s lemma]
-
A.
Siegel's theorem on integral points
Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
-
B.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
-
E.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Siegel’s lemma Target entity description: 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.
-
A.
Siegel's theorem on integral points
Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
-
B.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
-
E.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
- F. None of above. chosen
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
ⓘ
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
ⓘ
linked to:
Siegel’s lemma
absolute Siegel’s lemma ⓘ
linked to:
Siegel’s lemma
p-adic Siegel’s lemma ⓘ
linked to:
Siegel’s lemma
|
| 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 ⓘ |
| oftenExtendedTo | number fields ⓘ |
| oftenFormulatedOver | rational numbers ⓘ |
| provides | upper bounds on the size of integer solutions ⓘ |
| relatedTo |
Minkowski’s theorem
ⓘ
linked to:
Minkowski’s theorem on convex sets
Subspace theorem ⓘ Thue–Siegel–Roth theorem ⓘ |
| standardReference |
Bombieri and Gubler’s Heights in Diophantine Geometry
ⓘ
Cassels’ An Introduction to Diophantine Approximation ⓘ
linked to:
An Introduction to Diophantine Approximation
Serge Lang’s books on Diophantine approximation ⓘ
linked to:
An Introduction to Diophantine Approximation
|
| status | classical result in number theory ⓘ |
| subfield |
Diophantine approximation
ⓘ
geometry of numbers ⓘ |
| typicalBoundDependsOn |
maximal absolute value of coefficients
ⓘ
number of equations ⓘ number of variables ⓘ |
| 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 ⓘ |
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Input
Subject: Siegel’s lemma Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Siegel’s lemma
linked to: Siegel’s lemma
linked to: Siegel’s lemma