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ON THE NONARCHIMEDEAN METRIC MAHLER MEASURE
"... Abstract. Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a nonArchimedean version of the metric Mahler measure, denoted M∞, and prove that M∞(α) = 1 if and only if α is a root of ..."
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Abstract. Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a nonArchimedean version of the metric Mahler measure, denoted M∞, and prove that M∞(α) = 1 if and only if α is a root of unity. We further show that M ∞ defines a projective height on Q × /Tor(Q ×) as a vector space over Q. Finally, we demonstrate how to compute M∞(α) when α is a surd. 1.
ORTHOGONAL DECOMPOSITION OF THE SPACE OF ALGEBRAIC NUMBERS AND LEHMER’S PROBLEM
"... Abstract. We determine decompositions of the space of algebraic numbers modulo torsion by Galois field and degree which are orthogonal with respect to the natural inner product associated to the L 2 Weil height recently introduced by Allcock and Vaaler. Using these decompositions, we then introduce ..."
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Abstract. We determine decompositions of the space of algebraic numbers modulo torsion by Galois field and degree which are orthogonal with respect to the natural inner product associated to the L 2 Weil height recently introduced by Allcock and Vaaler. Using these decompositions, we then introduce vector space norms associated to the Mahler measure. We formulate L p Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p = 1 case and the SchinzelZassenhaus conjecture
Norms extremal with respect to the Mahler measure. Preprint available at http://arxiv.org/abs/1006.5503
"... Abstract. In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms o ..."
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Abstract. In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer’s conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the tmetric Mahler measures. 1.
A GENERALIZATION OF DIRICHLET’S SUNIT THEOREM
"... Abstract. We generalize Dirichlet’s Sunit theorem from the usual group of Sunits of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic Sunits modulo torsion is a Q ..."
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Abstract. We generalize Dirichlet’s Sunit theorem from the usual group of Sunits of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic Sunits modulo torsion is a Qvector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over Q retain their linear independence over R. 1.