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The Mahler measure of algebraic numbers: a survey.” Conference Proceedings
 University of Bristol
, 2008
"... Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though n ..."
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Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 17 (4 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Factoring bivariate sparse (lacunary) polynomials
 J. Complexity
"... Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Mo ..."
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Abstract. We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x,y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d. Moreover, we show that the factors over Q of degree ≤ d which are not binomials can also be computed in time polynomial in the sparse length of the input and in d.
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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Cited by 1 (1 self)
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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),
Problème de Lehmer pour les hypersurfaces de veriétés abéliennes de type C.M
"... [3] dans le cadre de l’intersection arithmétique, comment définir la hauteur des variétés projectives; l’idée étant de considérer un point comme une variété de dimension zéro et de généraliser ceci en dimension supérieure. ..."
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[3] dans le cadre de l’intersection arithmétique, comment définir la hauteur des variétés projectives; l’idée étant de considérer un point comme une variété de dimension zéro et de généraliser ceci en dimension supérieure.
On the Mahler measure in several variables
, 2008
"... If the total degree of a polynomial in n ≥ 2 variables of dimension n s bounded by a double exponential function in n, we show that its Mahler measure is bounded from below by an absolute constant > 1. ..."
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If the total degree of a polynomial in n ≥ 2 variables of dimension n s bounded by a double exponential function in n, we show that its Mahler measure is bounded from below by an absolute constant > 1.