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21
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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Cited by 39 (3 self)
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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.
A Lower Bound for the Canonical Height on Elliptic Curves over Abelian Extensions
 Duke Math. J
, 2003
"... Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K). ..."
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Cited by 25 (2 self)
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Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K).
A lower bound for the canonical height on abelian varieties over abelian extensions
, 2003
"... Let A be an abelian variety defined over a number field K and let ˆ h be the canonical height function on A ( ¯ K) attached to a symmetric ample line bundle L. We prove that there exists a constant C = C(A, K, L)> 0 such that ˆ h(P) ≥ C for all nontorsion points P ∈ A(K ab), where K ab is the ..."
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Cited by 9 (0 self)
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Let A be an abelian variety defined over a number field K and let ˆ h be the canonical height function on A ( ¯ K) attached to a symmetric ample line bundle L. We prove that there exists a constant C = C(A, K, L)> 0 such that ˆ h(P) ≥ C for all nontorsion points P ∈ A(K ab), where K ab is the maximal abelian extension of K.
Lower bounds for the canonical height on elliptic curves over abelian extensions
 IMRN
, 2002
"... Abstract. Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of nontorsion points on E defined over the maximal abelian extension K ab of K. This is analogous to results of AmorosoDvornici ..."
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Cited by 7 (1 self)
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Abstract. Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of nontorsion points on E defined over the maximal abelian extension K ab of K. This is analogous to results of AmorosoDvornicich and AmorosoZannier for the multiplicative group. We also show that if E has nonintegral jinvariant (so that in particular E does not have complex multiplication), then there exists C> 0 such that there are only finitely many points P ∈ E(K ab) of canonical height less than C. This strengthens a result of Hindry and Silverman. 1.
A uniform relative Dobrowolskis lower bound over abelian extensions.
, 2009
"... Abstract. Let L/K be an abelian extension of number fields. We prove an uniform lower bound for the height in L ∗ outside roots of unity. This lower bound depends only on the degree [L: K]. 1 Introduction. Let h be the Weil height on Q and let µ the set of roots of units. Let L be an abelian extensi ..."
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Cited by 1 (0 self)
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Abstract. Let L/K be an abelian extension of number fields. We prove an uniform lower bound for the height in L ∗ outside roots of unity. This lower bound depends only on the degree [L: K]. 1 Introduction. Let h be the Weil height on Q and let µ the set of roots of units. Let L be an abelian extension of the rational field. In a joint work with R. Dvornicich ([AmDv]) the first author proved that for any α ∈ L ∗ \µ h(α) ≥
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),
A Hitting Set Construction, with Applications to Arithmetic Circuit Lower Bounds
, 2009
"... Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential l ..."
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Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential lower bound for representations of polynomials such as n 2 i=1 (Xi − 1) under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the “hardness from derandomization” approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers. 1