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SMALL POINTS ON RATIONAL SUBVARIETIES OF TORI.
, 2009
"... In this article we study the distribution of the small points of proper subvarieties of the torus Gn m defined over Q. For n = 1, the problem corresponds to finding lower bounds for the Weil height of an algebraic number. Let α be a nonzero algebraic number of degree D which is not a root of unity. ..."
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In this article we study the distribution of the small points of proper subvarieties of the torus Gn m defined over Q. For n = 1, the problem corresponds to finding lower bounds for the Weil height of an algebraic number. Let α be a nonzero algebraic number of degree D which is not a root of unity. Lehmer (see [Leh 1933])
Bogomolov on tori revisited
"... Let V ⊆ Gnm ⊆ Pn be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ> 0 let V (θ) be the set of α ∈ V (Q) of Weil’s height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang), ..."
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Let V ⊆ Gnm ⊆ Pn be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ> 0 let V (θ) be the set of α ∈ V (Q) of Weil’s height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang),