Mahler measures of polynomials that are sums of a bounded number of monomials

Dobrowolski, Edward; Smyth, Chris

doi:10.1142/s1793042117500907

Cited by 6 publications

(3 citation statements)

References 14 publications

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“…We wish to establish a lower bound for M(P ) which depends on the coefficients and on the number of monomials, but which does not depend on the degree of P . Such a result was recently proved by Dobrowolski and Smyth [5]. We use a similar argument, but we obtain a sharper result that includes Mahler's inequality (1.3) as a special case.…”

Section: Introductionsupporting

confidence: 65%

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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Section: Introductionsupporting

confidence: 65%

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

Akhtari

Vaaler

2019

Int. J. Number Theory

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“…, whereas Dobrowolski and Smyth [23], as well as Akhtari and Vaaler [1], used the theorem of Lawton to study Mahler measures of polynomials with a bounded number of monomials. Finally, Dubickas [24] and Habegger [34] used Lawton's result in their investigations of sums of roots of unity, whereas, as we already mentioned, Smyth [58] used Lawton's limit formula to prove that the sets M(P ) defined in (1.1) are closed.…”

Section: Historical Remarksmentioning

confidence: 99%

Limits of Mahler measures in multiple variables

Brunault,

Guilloux,

Mehrabdollahei

et al. 2024

Annales De l'Institut Fourier

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We prove that certain sequences of Laurent polynomials, obtained from a fixed multivariate Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P . This generalises 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, extending 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.Résumé. -Nous prouvons que certaines suites de polynômes de Laurent, obtenues à partir d'un polynôme de Laurent P fixé par substitutions monomiales, donnent des suites de mesures de Mahler qui convergent vers la mesure de Mahler de P . Ce résultat généralisent des travaux antérieurs de Boyd et Lawton, qui considéraient des substitutions univariées. Nous obtenons aussi une borne supérieure explicite pour le terme d'erreur dans cette convergence, généralisant des travaux de Dimitrov et Habegger. Nous donnons enfin un développement asymptotique complet pour une famille particulière de polynômes bivariés, dont les mesures de Mahler avaient été étudiées indépendamment par la troisième autrice. d j=1 (z − α j ) for its complex factorisation, the product ∆ k (P ) = d j=1 (α k j − 1) is an integer for every k ∈ N. The case of the polynomial

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“…whereas Dobrowolski and Smyth [23], as well as Akhtari and Vaaler [1], used the theorem of Lawton to study Mahler measures of polynomials with a bounded number of monomials. Finally, Dubickas [24] and Habegger [34] used Lawton's result in their investigations of sums of roots of unity, whereas, as we already mentioned, Smyth [60] used Lawton's limit formula to prove that the sets M (P) defined in (1) are closed.…”

Section: Historical Remarksmentioning

confidence: 99%

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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Mahler measures of polynomials that are sums of a bounded number of monomials

Cited by 6 publications

References 14 publications

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

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

Limits of Mahler measures in multiple variables

Limits of Mahler measures in multiple variables

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