The mean values of logarithms of algebraic integers

Artūras Dubickas

Displaying similar documents to “The mean values of logarithms of algebraic integers”

A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

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Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$ . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

On the vectors associated with the roots of max-plus characteristic polynomials

Yuki Nishida, Sennosuke Watanabe, Yoshihide Watanabe (2020)

Applications of Mathematics

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We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation $A \otimes x = λ \otimes x$ for usual eigenvectors. Under a suitable...

The Lehmer constants of an annulus

Artūras Dubickas, Chris J. Smyth (2001)

Journal de théorie des nombres de Bordeaux

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Let $M (α)$ be the Mahler measure of an algebraic number $α$ , and $V$ be an open subset of $ℂ$ . Then its $L (V)$ is inf $M {(α)}^{1 / deg (α)}$ , the infimum being over all non-zero non-cyclotomic $α$ lying with its conjugates outside $V$ . We evaluate $L (V)$ when $V$ is any annulus centered at $0$ . We do the same for a variant of $L (V)$ , which we call the transfinite Lehmer constant $L_{\infty} (V)$ .Also, we prove the converse to Langevin’s Theorem, which states that $L (V) > 1$ if $V$ contains a point of modulus $1$ . We prove the corresponding result for...

The R₂ measure for totally positive algebraic integers

V. Flammang (2016)

Colloquium Mathematicae

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Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates $α ₁ = α, . . ., α_{d}$ are positive real numbers. We study the set ₂ of the quantities ${(\prod_{i = 1}^{d} {(1 + α ²_{i})}^{1 / 2})}^{1 / d}$ . We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.

The cubics which are differences of two conjugates of an algebraic integer

Toufik Zaimi (2005)

Journal de Théorie des Nombres de Bordeaux

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We show that a cubic algebraic integer over a number field $K,$ with zero trace is a difference of two conjugates over $K$ of an algebraic integer. We also prove that if $N$ is a normal cubic extension of the field of rational numbers, then every integer of $N$ with zero trace is a difference of two conjugates of an integer of $N$ if and only if the $3 -$ adic valuation of the discriminant of $N$ is not $4 .$

Cycles on algebraic models of smooth manifolds

Wojciech Kucharz (2009)

Journal of the European Mathematical Society

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Every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of $M$ . We study modulo 2 homology classes represented by algebraic subsets of $X$ , as $X$ runs through the class of all algebraic models of $M$ . Our main result concerns the case where $M$ is a spin manifold.

Distribution of nodes on algebraic curves in $ℂ^{N}$

Thomas Bloom, Norman Levenberg (2003)

Annales de l’institut Fourier

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Given an irreducible algebraic curves $A$ in $ℂ^{N}$ , let $m_{d}$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$ . Let $K$ be a nonpolar compact subset of $A$ , and for each $d = 1, 2, . . .,$ choose $m_{d}$ points ${A_{d j}}_{j = 1, . . ., m_{d}}$ in $K$ . Finally, let $Λ_{d}$ be the $d$ -th Lebesgue constant of the array ${A_{d j}}$ ; i.e., $Λ_{d}$ is the operator norm of the Lagrange interpolation operator $L_{d}$ acting on $C (K)$ , where $L_{d} (f)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${A_{d j}}_{j = 1, . . ., m_{d}}$ . Using techniques...

Polynomial relations amongst algebraic units of low measure

John Garza (2014)

Acta Arithmetica

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For an algebraic number field and a subset $α_{1}, . . ., α_{r} \subseteq_{}$ , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $ℚ [x_{1}, . . ., x_{r}]$ vanishing at the point $(α_{1}, . . ., α_{r})$ .

Algebraic Superconnections, $S U (2 ∣ 1)$ and Electroweak Interactions

R. Coquereaux (1992)

Recherche Coopérative sur Programme n°25

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Linear independence of linear forms in polylogarithms

Raffaele Marcovecchio (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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For $x \in ℂ$ , $| x | < 1$ , $s \in ℕ$ , let ${Li}_{s} (x)$ be the $s$ -th polylogarithm of $x$ . We prove that for any non-zero algebraic number $α$ such that $| α | < 1$ , the $ℚ (α)$ -vector space spanned by $1, {Li}_{1} (α), {Li}_{2} (α), \dots$ has infinite dimension. This result extends a previous one by Rivoal for rational $α$ . The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

Quasi-algebraic representability of sets in $R^{n}$ .

W. Waliszewski (1984)

Banach Center Publications

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Computation of linear algebraic equations with solvability verification over multi-agent networks

Xianlin Zeng, Kai Cao (2017)

Kybernetika

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In this paper, we consider the problem of solving a linear algebraic equation $A x = b$ in a distributed way by a multi-agent system with a solvability verification requirement. In the problem formulation, each agent knows a few columns of $A$ , different from the previous results with assuming that each agent knows a few rows of $A$ and $b$ . Then, a distributed continuous-time algorithm is proposed for solving the linear algebraic equation from a distributed constrained optimization viewpoint. The...