Results 1  10
of
58
Computing Heights on Elliptic Curves
, 1988
"... ] C.J. Smyth. On measures of polynomials in several variables. Bull. Australian Math. Soc., 23:4963, 1981. Corrigendum: G. Myerson and C.J. Smyth, 26 (1982), 317319. [soule1991] C. Soul'e. Geometrie d'Arakelov et th'eorie des nombres transcendants. Ast'erisque, 198200:355371, 1991. [stewart ..."
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Cited by 29 (3 self)
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] C.J. Smyth. On measures of polynomials in several variables. Bull. Australian Math. Soc., 23:4963, 1981. Corrigendum: G. Myerson and C.J. Smyth, 26 (1982), 317319. [soule1991] C. Soul'e. Geometrie d'Arakelov et th'eorie des nombres transcendants. Ast'erisque, 198200:355371, 1991. [stewart1977] C.L. Stewart. On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. Proceedings of the London Math. Soc., 35:425447, 1977. [stewart197778] C.L. Stewart. On a theorem of Kronecker and a related question of Lehmer. In S'eminaire de Th'eorie de Nombres Bordeaux 1977/78. Birkhauser, Basel, 1978. [stewart1978] C.L. Stewart. Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France, 106:169176, 1978. [szydlo1985] B. Szydlo. An application of some theorems of G. Szegoe to Mahler measure of polynomials. Discuss. Math., 7:145148, 1985. [tatethesis]
Some examples of Mahler measures as multiple polylogarithms
 J. Number Theory
"... The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial. Key words: Mahler measure, Lfunctions, polylogarithms, hyperlogarithms ..."
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Cited by 23 (15 self)
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The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial. Key words: Mahler measure, Lfunctions, polylogarithms, hyperlogarithms, polynomials, Jensen’s formula
An explicit formula for the Mahler measure of a family of 3variable polynomials
"... An explicit formula for the Mahler measure of the 3variable Laurent polynomial a + bx 1 + cy +(a + bx + cy)z is given, in terms of dilogarithms and trilogarithms. 2000 Mathematics Subject Classification. 11R06. ..."
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Cited by 20 (0 self)
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An explicit formula for the Mahler measure of the 3variable Laurent polynomial a + bx 1 + cy +(a + bx + cy)z is given, in terms of dilogarithms and trilogarithms. 2000 Mathematics Subject Classification. 11R06.
New 5F4 hypergeometric transformations, threevariable Mahler measures, and formulas for 1/π
 Ramanujan J
"... New relations are established between families of threevariable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1 ..."
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Cited by 18 (7 self)
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New relations are established between families of threevariable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1
The Mahler measure of algebraic numbers: a survey.” Conference Proceedings
 University of Bristol
, 2008
"... Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though n ..."
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Cited by 17 (2 self)
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Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.
Mahler measure of some nvariable polynomial families, (preprint 2004
"... The Mahler measures of some nvariable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet Lseries, generalizing the results of [13]. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric funct ..."
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Cited by 15 (10 self)
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The Mahler measures of some nvariable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet Lseries, generalizing the results of [13]. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions. Keywords Mahler measure, Riemann zeta function, Lfunctions, polynomials, Bernoulli numbers, symmetric functions
Hypergeometric formulas for lattice sums and Mahler measures
, 2010
"... We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd’s conjectured identities between Mahler measures and special values of Lseries of elliptic curves. 1 ..."
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Cited by 14 (9 self)
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We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd’s conjectured identities between Mahler measures and special values of Lseries of elliptic curves. 1
The arithmetic and geometry of Salem numbers
 Bull. Amer. Math. Soc
, 1991
"... Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and ar ..."
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Cited by 13 (2 self)
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Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1. Contents
Functional equations for Mahler measures of genusone curves
 ALGEBRA AND NUMBER THEORY
"... In this paper we will establish functional equations for Mahler measures of families of genusone twovariable polynomials. These families were previously studied by Beauville [3], and their Mahler measures were considered by Boyd [11] and RodriguezVillegas [19]. Bertin [8], Zagier [26], and Stiens ..."
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Cited by 12 (11 self)
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In this paper we will establish functional equations for Mahler measures of families of genusone twovariable polynomials. These families were previously studied by Beauville [3], and their Mahler measures were considered by Boyd [11] and RodriguezVillegas [19]. Bertin [8], Zagier [26], and Stienstra [24]. Our functional equations allow us to prove identities between Mahler measures that were conjectured by Boyd. As a corollary, we also establish some new transformations for hypergeometric functions.
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 10 (3 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.