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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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 ⓘ

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Enrico Bombieri → knownFor → Bombieri–Pila determinant method ⓘ