277 research outputs found

    Root separation for irreducible integer polynomials

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    We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).Comment: 8 pages; revised version; to appear in Bull. Lond. Math. So

    On the Littlewood conjecture in fields of power series

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    Let \k be an arbitrary field. For any fixed badly approximable power series Θ\Theta in \k((X^{-1})), we give an explicit construction of continuum many badly approximable power series Φ\Phi for which the pair (Θ,Φ)(\Theta, \Phi) satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series

    On the complexity of algebraic number I. Expansions in integer bases

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    Let b≥2b \ge 2 be an integer. We prove that the bb-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion

    On the Maillet--Baker continued fractions

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    We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number
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