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Heights of roots of polynomials with odd coefficients

J. Garza, M. I. M. Ishak, M. J. Mossinghoff, C. G. Pinner, B. Wiles (2010)

Journal de Théorie des Nombres de Bordeaux

Let α be a zero of a polynomial of degree n with odd coefficients, with α not a root of unity. We show that the height of α satisfies h ( α ) 0 . 4278 n + 1 . More generally, we obtain bounds when the coefficients are all congruent to 1 modulo m for some m 2 .

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let A m be the mth coefficient of the square f(x)² of a unimodular...

Higher Mahler measure of an n-variable family

Matilde N. Lalín, Jean-Sébastien Lechasseur (2016)

Acta Arithmetica

We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity.

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