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16
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.
Small points on subvarieties of a torus
 Journal
, 2009
"... Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. Especially, we determine whether such a set is or not dense in V. We the ..."
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Cited by 21 (3 self)
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Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. Especially, we determine whether such a set is or not dense in V. We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees. As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor. 1.
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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Cited by 13 (2 self)
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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.
The intersection of a curve with a union of translated codimension 2 subgroups in a power of an elliptic curve
 Algebra and Number Theory
"... Abstract. Let E be an elliptic curve. Consider an irreducible algebraic curve C embedded in E g. The curve is transverse if it is not contained in any translate of a proper algebraic subgroup of E g. Furthermore C is weaktransverse if it is not contained in any proper algebraic subgroup. Suppose th ..."
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Abstract. Let E be an elliptic curve. Consider an irreducible algebraic curve C embedded in E g. The curve is transverse if it is not contained in any translate of a proper algebraic subgroup of E g. Furthermore C is weaktransverse if it is not contained in any proper algebraic subgroup. Suppose that both E and C are defined over the algebraic numbers. We prove that the algebraic points of a transverse curve C which are close to the union of all algebraic subgroups of E g of codimension 2 translated by points in a subgroup Γ of E g of finite rank are a set of bounded height. The notion of close is defined using a height function. If Γ is trivial, it is sufficient to suppose that C is weaktransverse. Then, we introduce a method to determine the finiteness of these sets. From a conjectural lower bound for the normalised height of a transverse curve C, we deduce that the above sets are finite. At present, such a lower bound exists for g≤3. Our results are optimal, for what concerns the codimension of the algebraic
Nondense subsets of varieties in a power of an elliptic curve
"... Let E be an elliptic curve without C.M. defined over Q. We show that on a transverse ddimensional variety V ⊂ Eg, the set of algebraic points of bounded height which are close to the union of all algebraic subgroups of Eg of codimension d + 1 translated by a point in a subgroup Γ of Eg of finite ra ..."
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Let E be an elliptic curve without C.M. defined over Q. We show that on a transverse ddimensional variety V ⊂ Eg, the set of algebraic points of bounded height which are close to the union of all algebraic subgroups of Eg of codimension d + 1 translated by a point in a subgroup Γ of Eg of finite rank, is nonZariski dense in V. The notion of close is defined using a height function. If Γ = 0, it is sufficient to assume that V is weaktransverse. This result is optimal with respect to the codimension of the algebraic subgroups. The method is based on an essentially optimal effective version of the Bogomolov Conjecture. Such an effective result is proven for subvarieties of Eg. If we assume that the sets have bounded height, then we can prove that they are not Zariski dense. A conjecture, known in some special cases, claims that the sets in question have bounded height. We prove here a new case. In conclusion, our results prove a generalized case of a conjecture by Zilber and by Pink in Eg. 1. introduction In this article all algebraic varieties are defined over Q and we consider only
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])
Geometric lower bounds for the normalized height of hypersurfaces
, 2006
"... We are here concerned in the Bogomolov’s problem for the hypersurfaces; we give a geometric lower bound for the height of a hypersurface of G n m (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of G n m. This is an analogue of ..."
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We are here concerned in the Bogomolov’s problem for the hypersurfaces; we give a geometric lower bound for the height of a hypersurface of G n m (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of G n m. This is an analogue of a result of F. Amoroso and S. David who give a lower bound for the height of nontorsion hypersurfaces defined and irreducible over the rationals.
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),
NONDENSE SUBSETS OF ALGEBRAIC POINTS ON A VARIETY
, 2008
"... We prove a new generalized case of the so called ZilberPink Conjecture. Let E be an elliptic curve without C.M. defined over Q. We introduce a method to show that the set of algebraic points on a transverse ddimensional variety V ⊂ Eg, which are close to the union of all algebraic subgroups of Eg ..."
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We prove a new generalized case of the so called ZilberPink Conjecture. Let E be an elliptic curve without C.M. defined over Q. We introduce a method to show that the set of algebraic points on a transverse ddimensional variety V ⊂ Eg, which are close to the union of all algebraic subgroups of Eg of codimension d + 1 translated by a point in a subgroup Γ of Eg of finite rank, is nonZariski dense in V. The notion of close is defined using a height function. If Γ = 0, it is sufficient to assume that V is weaktransverse. This result is optimal with respect to the codimension of the algebraic subgroups. The method basis on an essentially optimal Bogomolov type bound as given in Conjecture 1.2. Such a bound is proven for d ≥ g −2. Also, we use the boundedness of the height of the set in question. Such a bound is known is some cases; we prove here a new case.