The Mahler measure of algebraic numbers: a survey

Smyth, Chris

doi:10.1017/cbo9780511721274.021

Cited by 94 publications

(83 citation statements)

References 123 publications

(117 reference statements)

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“…Checking all n with 0 < |n| ≤ 200 we find that B(f, n) = +1 and B(f, n) = −1 for ±n = 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 17, 18, 21, 23, 27, 29, 34, 37, 47, 63, 65, 74. (This polynomial, famous for having the minimal known Mahler measure greater than 1, was found by D.H. Lehmer [11]; see [16] for more on this. )…”

Section: Examples 4 Andmentioning

confidence: 86%

Unimodular integer circulants

Cremona¹

2008

Math. Comp.

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Abstract. We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial f (x) ∈ Z[x], determine all those n ∈ N such that Res(f (x), x n − 1) = ±1. In this paper we describe methods for resolving this problem, including a method based on the use of Strassman's Theorem on p-adic power series, which are effective in many cases. The methods are illustrated with examples arising in the study of cyclically presented groups and further examples which illustrate the strengths and weaknesses of the methods for polynomials of higher degree.

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Section: Examples 4 Andmentioning

confidence: 86%

Unimodular integer circulants

Cremona¹

2008

Math. Comp.

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“…Note that the trivial upper bound M (P ) < ||P || 1 = n + 1 for all P ∈ LP n follows from the monotonicity of L-norms and Proposition 2.1. We remark that the computation of the exact values of the Mahler measure of polynomials is quite difficult; see, for instance, a survey by C. Smyth [14]. As a first step towards Conjecture 2.4 we prove the inequality for the Mahler measures of polynomials R n : Theorem 2.6.…”

Section: Conjecture 25 the Mahler Measures Of Polynomialsmentioning

confidence: 90%

On a class of polynomials related to Barker sequences

Borwein

Choi

Jankauskas

2012

Proc. Amer. Math. Soc.

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Abstract. For an odd integer n > 0, we introduce the class LP n of Laurent polynomialswith all coefficients c k equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials P ∈ LP n have large Mahler measures, namely, M (P ) > (n + 1)/2. We conjecture that minimal Mahler measures in the class LP n are attained by the polynomials R n (z) and R n (−z), whereis a polynomial with all the coefficients c k = 1. We prove thatThe results of experimental computations on polynomials in the class LP n suggest two conjectures which could shed light on the long-standing Barker problem.

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“…Boyd observes that there appears to be a tight connection to K-theory. An early result due to Smyth (see [51], also [53]) is that…”

Section: Example 3 (Periods and Mahler Measures [39])mentioning

confidence: 99%