Bombieri–Pila determinant method
E571014
The Bombieri–Pila determinant method is a technique in analytic and Diophantine geometry used to obtain upper bounds on the number of rational or integral points of bounded height lying on algebraic curves or more general sets.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bombieri–Pila determinant method canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6150019 — 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: Bombieri–Pila determinant method Context triple: [Enrico Bombieri, knownFor, Bombieri–Pila determinant method]
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A.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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B.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
-
C.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
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D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
- 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: Bombieri–Pila determinant method Target entity description: The Bombieri–Pila determinant method is a technique in analytic and Diophantine geometry used to obtain upper bounds on the number of rational or integral points of bounded height lying on algebraic curves or more general sets.
-
A.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
B.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
-
C.
Hodge–Riemann bilinear relations
The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical method
ⓘ
technique in Diophantine geometry ⓘ technique in analytic number theory ⓘ |
| appliesOver |
integers
ⓘ
rational numbers ⓘ real numbers ⓘ |
| appliesTo |
algebraic curves
ⓘ
real analytic sets ⓘ sets definable in o-minimal structures ⓘ transcendental curves ⓘ |
| assumption |
curve has bounded degree
ⓘ
points have bounded height ⓘ |
| field |
Diophantine geometry
NERFINISHED
ⓘ
analytic number theory ⓘ arithmetic geometry ⓘ |
| goal |
bound number of integral points of bounded height
ⓘ
bound number of rational points of bounded height ⓘ |
| historicalContext | developed in late 20th century ⓘ |
| inspired | later determinant methods in Diophantine geometry ⓘ |
| involves |
balancing degree of polynomials and number of points
ⓘ
construction of auxiliary polynomials vanishing on many points ⓘ estimating determinants built from point coordinates ⓘ |
| namedAfter |
Enrico Bombieri
NERFINISHED
ⓘ
János Pintz Pila NERFINISHED ⓘ |
| output |
upper bound for number of integral points
ⓘ
upper bound for number of rational points ⓘ |
| relatedTo |
Bombieri–Pila theorem
NERFINISHED
ⓘ
Pila–Wilkie theorem NERFINISHED ⓘ Pila–Zannier method NERFINISHED ⓘ determinant method of Heath-Brown NERFINISHED ⓘ |
| techniqueType |
determinant-based counting method
ⓘ
interpolation method ⓘ |
| typicalBoundForm | subpolynomial in the height parameter for transcendental curves ⓘ |
| typicalInput |
algebraic variety
ⓘ
curve in the plane ⓘ |
| usedFor |
counting rational points on transcendental sets
ⓘ
quantitative results on rational points ⓘ unlikely intersections problems ⓘ |
| usesConcept |
auxiliary polynomial
ⓘ
determinant ⓘ height of rational points ⓘ interpolation determinant ⓘ lattice point counting ⓘ |
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
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Input
Subject: Bombieri–Pila determinant method Description of subject: The Bombieri–Pila determinant method is a technique in analytic and Diophantine geometry used to obtain upper bounds on the number of rational or integral points of bounded height lying on algebraic curves or more general sets.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.