Bombieri–Pila determinant method

E571014

mathematical method technique in Diophantine geometry technique in analytic number theory

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.

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All labels observed (3)

Label	Occurrences
Bombieri–Pila determinant method canonical	1
Bombieri–Pila theorem	1
determinant method of Heath-Brown	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]

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
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 ⓘ 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 ⓘ János Pintz Pila ⓘ
output	upper bound for number of integral points ⓘ upper bound for number of rational points ⓘ
relatedTo	Bombieri–Pila theorem ⓘ linked to: Bombieri–Pila determinant method Pila–Wilkie theorem ⓘ Pila–Zannier method ⓘ determinant method of Heath-Brown ⓘ linked to: Bombieri–Pila determinant method
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

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Enrico Bombieri → knownFor → Bombieri–Pila determinant method ⓘ

Bombieri–Pila determinant method → relatedTo → determinant method of Heath-Brown ⓘ

linked to: Bombieri–Pila determinant method

Bombieri–Pila determinant method → relatedTo → Bombieri–Pila theorem ⓘ