geometry of numbers
E637302
Geometry of numbers is a branch of number theory that studies the properties of integers and Diophantine equations using the geometry of lattices and convex bodies in Euclidean space.
All labels observed (1)
| Label | Occurrences |
|---|---|
| geometry of numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030794 — 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: geometry of numbers Context triple: [Diophantine approximation, relatedTo, geometry of numbers]
-
A.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
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B.
Diophantine geometry
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
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C.
Hermite constant
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
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D.
Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
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E.
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.
- 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: geometry of numbers Target entity description: Geometry of numbers is a branch of number theory that studies the properties of integers and Diophantine equations using the geometry of lattices and convex bodies in Euclidean space.
-
A.
Diophantine approximation
Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
-
B.
Diophantine geometry
Diophantine geometry is the branch of number theory that studies solutions to polynomial equations with integer or rational coefficients using geometric methods, particularly those from algebraic geometry.
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C.
Hermite constant
The Hermite constant is a number in each dimension that measures the densest possible lattice sphere packing, playing a central role in the geometry of numbers and lattice theory.
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D.
Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
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E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf | branch of mathematics ⓘ |
| appliesTo |
Diophantine approximation
NERFINISHED
ⓘ
Diophantine inequalities ⓘ algebraic number theory ⓘ lattice point counting problems ⓘ quadratic forms ⓘ sphere packing problems ⓘ transference principles ⓘ |
| centralConcept |
Minkowski sum
NERFINISHED
ⓘ
convex symmetric body ⓘ covering radius ⓘ lattice in Euclidean space ⓘ packing density ⓘ reduction theory of quadratic forms ⓘ successive minima ⓘ |
| developedIn |
early 20th century
ⓘ
late 19th century ⓘ |
| field | number theory ⓘ |
| hasApplicationIn |
coding theory
ⓘ
cryptography ⓘ discrete tomography ⓘ optimization ⓘ |
| hasMethod |
lattice point enumeration in convex bodies
ⓘ
reduction of lattices ⓘ successive minima estimates ⓘ volume comparison arguments ⓘ |
| hasTheorem |
Blichfeldt theorem
NERFINISHED
ⓘ
Hermite constant bounds NERFINISHED ⓘ Mahler compactness theorem NERFINISHED ⓘ Minkowski convex body theorem NERFINISHED ⓘ Minkowski lattice point theorem NERFINISHED ⓘ Minkowski linear forms theorem NERFINISHED ⓘ Siegel mean value theorem NERFINISHED ⓘ |
| historicalFigure |
Carl Ludwig Siegel
NERFINISHED
ⓘ
Hermann Minkowski NERFINISHED ⓘ Kurt Mahler NERFINISHED ⓘ Louis Mordell NERFINISHED ⓘ |
| introducedBy | Hermann Minkowski NERFINISHED ⓘ |
| relatedTo |
algebraic geometry
ⓘ
arithmetic geometry ⓘ discrete geometry ⓘ functional analysis ⓘ metric number theory ⓘ |
| studies |
Diophantine equations
ⓘ
convex bodies ⓘ integer points in Euclidean space ⓘ lattices ⓘ |
| uses |
Euclidean geometry
ⓘ
convex geometry ⓘ lattice theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: geometry of numbers Description of subject: Geometry of numbers is a branch of number theory that studies the properties of integers and Diophantine equations using the geometry of lattices and convex bodies in Euclidean space.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.