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The Approximation of Solutions to Linear Homogeneous Differential Equations by Rational Functions.

Charles Osgood (1980)

Monatshefte für Mathematik

The EM Algorithm applied to determining new limit points of Mahler measures

Souad El Otmani, Georges Rhin, Jean-Marc Sac-Épée (2010)

Control and Cybernetics

The factorization of certain quadrinominals.

W.H. Mills (1985)

Mathematica Scandinavica

The integer transfinite diameter of intervals and totally real algebraic integers

V. Flammang, G. Rhin, C. J. Smyth (1997)

Journal de théorie des nombres de Bordeaux

In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals $[\frac{r}{s}, \frac{r}{s} + δ]$ , where $\frac{r}{s}$ is a fixed rational and $δ \to 0$ . We also study functions $g_{-}, g, g^{+}$ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical polynomials, associated to a given interval $I$ . We show how to estimate from below the proportion of roots of an integer polynomial which is sufficiently...

The Lehmer constants of an annulus

Artūras Dubickas, Chris J. Smyth (2001)

Journal de théorie des nombres de Bordeaux

Let $M (α)$ be the Mahler measure of an algebraic number $α$ , and $V$ be an open subset of $ℂ$ . Then its Lehmer constant $L (V)$ 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 $L_{\infty} (V)$ .