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## Effective results for points on certain subvarieties of tori (2008)

### Citations

81 |
Positive line bundles on arithmetic varieties
- Zhang
- 1995
(Show Context)
Citation Context ... · · · xTHT (1·1) where xi ∈ ε, Hi is an algebraic subgroup of (Q∗)N and xiHi ⊂ X for i = 1, . . . , T . This encompasses earlier work of Liardet [9] and Laurent [8] (who considered X) and Zhang =-=[17]-=- (who considered X {x ∈ (Q∗)N : h(x) < ε}). Bombieri and Zannier [4] and Schmidt [15] proved precise quantitative versions for Zhang’s result with an explicit positive value for ε and an explicit up... |

49 | Minorations des hauteurs normalisées des sous-variétés des tores - Philippon - 1999 |

49 |
Equations diophantiennes exponentielles
- Laurent
- 1984
(Show Context)
Citation Context ...a finite union of translates x1H1 · · · xTHT (1·1) where xi ∈ ε, Hi is an algebraic subgroup of (Q∗)N and xiHi ⊂ X for i = 1, . . . , T . This encompasses earlier work of Liardet [9] and Laurent =-=[8]-=- (who considered X) and Zhang [17] (who considered X {x ∈ (Q∗)N : h(x) < ε}). Bombieri and Zannier [4] and Schmidt [15] proved precise quantitative versions for Zhang’s result with an explicit pos... |

34 | An effective lower bound for the height of algebraic numbers
- Voutier
- 1996
(Show Context)
Citation Context ...ose that α is a non-zero algebraic number of degree d, which is not a root of unity. Then h(α) c(d)−1 where c(1) = 1 log 2 , c(d) = d(log 3d) 3 2 if d 2. Proof. This is the main result of Voutier =-=[16]-=-. LEMMA 4·4. (i) Let α, β ∈ Q∗. Then there are at most two points x = (x, y) ∈ (Q∗)2 such that αx + βy = 1, h(x) 0.03. (ii) Let f (X, Y ) ∈ Q[X, Y ] be an irreducible polynomial which is not a binom... |

19 | Mordell-Lang plus Bogomolov
- Poonen
- 1998
(Show Context)
Citation Context ...bvariety that is a subgroup of (Q∗)N , and by a translate of an algebraic subgroup a coset xH = {x·y : y ∈ H}, where H is an algebraic subgroup of (Q∗)N and x ∈ (Q∗)N . It follows from work of Poonen =-=[12]-=- that there is ε > 0 depending only on N and the degree of X, such that X ε is contained in a finite union of translates x1H1 · · · xTHT (1·1) where xi ∈ ε, Hi is an algebraic subgroup of (Q∗)... |

14 |
Sur les sous-variétés des tores
- Rémond
(Show Context)
Citation Context ... positive value for ε and an explicit upper bound for the number T of translates, both depending only on N and the degree of X and their result was further improved by various authors. Later, Rémond =-=[13]-=- proved a quantitative version of Poonen’s result with an explicit positive value for ε depending on N and the degree of X and an explicit upper bound for T depending only on N , the degree of X and t... |

12 | Lower bounds of heights of points on hypersurfaces - Beukers, Zagier - 1997 |

8 |
Treybig. A sharp bound for solutions of linear Diophantine equations
- Borosh, Flahive, et al.
- 1989
(Show Context)
Citation Context ...he equation xU = m in x ∈ Zr (7·4) has a solution. Then equation (7·4) has a solution x0 ∈ Zr such that ‖x0‖∞ kk/2V k−1 max(V, R). Proof. According to a result of Borosh, Flahive, Rubin and Treybig =-=[5]-=-, (7·4) has a solution x0 with ‖x0‖∞ W , where W is the maximum of the absolute values of the minors of the augmented matrix with U on the first r rows and m on the last row. Now our Lemma follows e... |

7 |
Algebraic points on subvarieties of Gnm
- Bombieri, Zannier
- 1995
(Show Context)
Citation Context ... and xiHi ⊂ X for i = 1, . . . , T . This encompasses earlier work of Liardet [9] and Laurent [8] (who considered X) and Zhang [17] (who considered X {x ∈ (Q∗)N : h(x) < ε}). Bombieri and Zannier =-=[4]-=- and Schmidt [15] proved precise quantitative versions for Zhang’s result with an explicit positive value for ε and an explicit upper bound for the number T of translates, both depending only on N and... |

5 | Effective results for linear equations in two unknowns from a multiplicative division group,
- B\’erczes, Evertse, et al.
- 2009
(Show Context)
Citation Context ...d 2·5] by giving explicit bounds for the heights of the points x contained both in C and in , ε or C(, ε), respectively. Our proofs are based on a new Diophantine approximation theorem obtained in =-=[1]-=- (see Lemma 4·1 in Section 4 below). Secondly, we consider varieties in (Q∗)N given by equations f1(x) = 0, . . . , fm(x) = 0 where each polynomial fi is a binomial or trinomial. Here we apply effecti... |

5 |
on subvarieties of tori, in A Panorama of Number Theory or the View From Baker’s Garden (Zürich
- Evertse, Points
- 1999
(Show Context)
Citation Context ...exc to be the set of x ∈ X with the property that there exists an algebraic subgroup H of (Q∗)N of dimension > 0 such that xH ⊂ X, and let X0 := X \ Xexc. The second author stated in the survey paper =-=[7]-=- that there exists ε > 0 depending on N , X and such that X0 C(, ε) is finite. This was proved in a more general form by Rémond [13]. In the case that X is a curve, Rémond gave, for some explic... |

5 | Lower bounds for heights on finitely generated groups - SCHLICKEWEI - 1997 |

4 |
Sur une conjecture de Serge Lang. Astérisque 24–25
- LIARDET
- 1975
(Show Context)
Citation Context ...is contained in a finite union of translates x1H1 · · · xTHT (1·1) where xi ∈ ε, Hi is an algebraic subgroup of (Q∗)N and xiHi ⊂ X for i = 1, . . . , T . This encompasses earlier work of Liardet =-=[9]-=- and Laurent [8] (who considered X) and Zhang [17] (who considered X {x ∈ (Q∗)N : h(x) < ε}). Bombieri and Zannier [4] and Schmidt [15] proved precise quantitative versions for Zhang’s result with... |

3 |
Heights of points on subvarieties of Gnm
- Schmidt
- 1996
(Show Context)
Citation Context ...r i = 1, . . . , T . This encompasses earlier work of Liardet [9] and Laurent [8] (who considered X) and Zhang [17] (who considered X {x ∈ (Q∗)N : h(x) < ε}). Bombieri and Zannier [4] and Schmidt =-=[15]-=- proved precise quantitative versions for Zhang’s result with an explicit positive value for ε and an explicit upper bound for the number T of translates, both depending only on N and the degree of X ... |

2 | Petits points d’une surface - PONTREAU |

2 |
A Mordell–Lang plus Bogolomov type result for curves in G2m
- PONTREAU
(Show Context)
Citation Context ...e explicit value of ε depending on N , the rank of and the height and degree of X, an explicit upper bound for the cardinality of X0 C(, ε); his result was recently improved by the fourth author =-=[11]-=- for curves in (Q ∗ )2. For higher dimensional varieties, such a quantitative version has as yet not been established. The purpose of this paper is to derive, for certain special classes of varieties ... |