A Generalization of a Theorem of Boyd and Lawton

Issa, Zahraa; Laĺın, Matilde

doi:10.4153/cmb-2012-010-2

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…Furthermore, Lalín and Sinha [39] mention a generalization of Lawton's theorem [39,Theorem 30] to the multiple Mahler measure, introduced in previous work of Kurokawa, Lalín and Ochiai [38]. Such a generalization was rigorously proved by Issa and Lalín [36], who dealt also with the generalized Mahler measures defined by Gon and Oyanagi [29]. On the other hand, Carter, Lalín, Manes, Miller and Mocz [14, Proposition 1.3] recently proved a weak generalization of Lawton's result to dynamical Mahler measures (introduced in [14, Definition 1.1]).…”

Section: Historical Remarksmentioning

confidence: 99%

“…Note in particular that the exponent appearing in this bound is halved with respect to the ones featured in (35), as a consequence of the square root appearing in the function ψ(s) := ε/(e 2s − 1). However, Lemma 4.12 allows us to have a final bound (36) featuring the same exponent appearing in (35). More precisely, Lemma 4.12 allows us to define δ ε (P) using √ ε instead of ε.…”

Section: Lemma 413 Let P ∈ C[z ±1mentioning

confidence: 99%

“…This in turn allows us to set ε := (diam(P) • L 1 (P) • a • log(ρ(A))/ρ(A)) 2 . Then, we can bound the last term appearing in (26) by applying Corollary 4.14, and the presence of the square in the definition of ε implies that the exponent appearing in (36) can be taken to be the same as the one featured in (35).…”

Section: Lemma 413 Let P ∈ C[z ±1mentioning

confidence: 99%

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Limits of Mahler measures in multiple variables

Brunault¹,

Guilloux²,

Mehrabdollahei³

et al. 2022

Preprint

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We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

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Section: Historical Remarksmentioning

confidence: 99%

Section: Lemma 413 Let P ∈ C[z ±1mentioning

confidence: 99%

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Limits of Mahler measures in multiple variables

Brunault¹,

Guilloux²,

Mehrabdollahei³

et al. 2022

Preprint

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“…To complete the proof we appeal to an inequality of Boyd [2, Lemma 2], which asserts that if b is a parameter in Z M then The proposed identity (4.8) was later verified by Lawton [8] (see also [4] and [7]).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Lower bounds for Mahler measure that depend on the number of monomials

Akhtari

Vaaler

2019

Int. J. Number Theory

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We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of Mahler. In M variables our result depends on Z M as an ordered group, and in general our lower bound depends on the choice of ordering. P (z) = (z ± 1) N , 2010 Mathematics Subject Classification. 11R06.

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“…Generalized Mahler measures were introduced by Gon and Oyanagi [GO04] who studied their basic properties, computed some examples, and related them to multiple sine functions and special values of Dirichlet L-functions. They were also studied in [IL13,Lal08].…”

Section: The Areal Versions Of Generalized Higher and Zeta Mahler Mea...mentioning

confidence: 99%