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The Absolute Subspace Theorem and linear equations with unknowns from a multiplicative group
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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),
Effective results for points on certain subvarieties of tori
, 2008
"... The combined conjecture of LangBogomolov for tori gives an accurate description of the set of those points x of a given subvariety X of GNm(Q) = (Q)N, that with respect to the height are “very close ” to a given subgroup of finite rank of GNm(Q). Thanks to work of Laurent, Poonen and Bogomolov, t ..."
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The combined conjecture of LangBogomolov for tori gives an accurate description of the set of those points x of a given subvariety X of GNm(Q) = (Q)N, that with respect to the height are “very close ” to a given subgroup of finite rank of GNm(Q). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties X, effective versions of the LangBogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Bakertype logarithmic forms estimates and Bogomolovtype estimates for the number of points on the variety X with very small height. 1.
Ultrarigid periodic frameworks
, 2014
"... We give an algebraic characterization of when a ddimensional periodic framework has no nontrivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable, and we provide a simple algorithm that does not require complicated algebraic computations. In dimensio ..."
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We give an algebraic characterization of when a ddimensional periodic framework has no nontrivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable, and we provide a simple algorithm that does not require complicated algebraic computations. In dimension d = 2, we give a combinatorial characterization in the special case when the the number of edge orbits is the minimum possible for ultrarigidity. All our results apply to a fully flexible, fixed area, or fixed periodicity lattice.