Diophantine Geometry

Hindry, Marc; Silverman, Joseph H.

doi:10.1007/978-1-4612-1210-2

Cited by 410 publications

(370 citation statements)

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“…We shortly explain how this result is deduced from Proposition 6·1. If f (x, y) = 0 then there exists an absolutely irreducible factor g(X, Y ) of f , which then fulfils the conditions of Proposition 6·1, thus we may apply that for g. Further, since g divides f it is also well known that h(g) ≤ h(f ) + 2N (see Proposition B.7.3 of [19]). …”

Section: ·1 the Results For The Number Field Casementioning

confidence: 94%

Effective results for unit points on curves over finitely generated domains

Bérczes

2015

Math. Proc. Camb. Phil. Soc.

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Let A be a commutative domain of characteristic 0 which is finitely generated over Z as a Z-algebra. Denote by A * the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the setwhere F (X, Y ) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form X m Y n − α or X m − αY n , with m, n ∈ Z ≥0 , max(m, n) > 0, α ∈ K * . This result is a common generalization of effective results of Evertse and Győry (2013) on S-unit equations over finitely generated domains, of Bombieri and Gubler (2006) on the equation F (x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem (1960) on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.

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Section: ·1 the Results For The Number Field Casementioning

confidence: 94%

Effective results for unit points on curves over finitely generated domains

Bérczes

2015

Math. Proc. Camb. Phil. Soc.

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“…Proposition 8. 8. Let X be an irreducible variety of dimension n, L an ample line bundle on X, and x ∈ X(k).…”

Section: David Mckinnon and Mike Rothmentioning

confidence: 99%

Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

McKinnon

Roth

2014

Invent. math.

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Abstract. In this paper, we associate an invariant α x (L) to an algebraic point x on an algebraic variety X with an ample line bundle L. The invariant α measures how well x can be approximated by rational points on X, with respect to the height function associated to L. We show that this invariant is closely related to the Seshadri constant ǫ x (L) measuring local positivity of L at x, and in particular that Roth's theorem on P 1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

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Section: Rational Points: Beyond 2/dmentioning

confidence: 99%

“…Note that for x ∈ C(K) (5.1) ||j a (x)|| 2 Θ = gh a (x) + O( 1 + h a (x)) (this follows from [HS,Theorem B.5.9], since j * a Θ is algebraically equivalent to g · (a); cf. [HS,Theorem A.8.2.1]). By the proof of Mumford's gap principle (see [HS,Theorems A.8.2.1 and B.6.5]), one has, for any x, y ∈ C(K), Now suppose that x, y ∈ C(K) are chosen so that h a (x), h a (y) ≤ h 0 .…”

Section: Rational Points: Beyond 2/dmentioning

confidence: 99%

“…[HS,Theorem A.8.2.1]). By the proof of Mumford's gap principle (see [HS,Theorems A.8.2.1 and B.6.5]), one has, for any x, y ∈ C(K), Now suppose that x, y ∈ C(K) are chosen so that h a (x), h a (y) ≤ h 0 . The theory of local heights ([Se2]) shows that, if x = y ∈ C(K) reduce to the same point modulo p, any prime ideal of O K , then h ∆ (x, y) ≥ log(N p) + O(1).…”

Section: Rational Points: Beyond 2/dmentioning

confidence: 99%

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2006

J. Amer. Math. Soc.

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Diophantine Geometry

Cited by 410 publications

References 0 publications

Effective results for unit points on curves over finitely generated domains

Effective results for unit points on curves over finitely generated domains

Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

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