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Journal articles on the topic 'Mahler measure'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 January 2023

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1

Issa, Zahraa, and Matilde Lalín. "A Generalization of a Theorem of Boyd and Lawton." Canadian Mathematical Bulletin 56, no. 4 (December 1, 2013): 759–68. http://dx.doi.org/10.4153/cmb-2012-010-2.

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Abstract.The Mahler measure of a nonzero n-variable polynomial P is the integral of log |P| on the unit n-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of log |P| for possibly different P’s), multiple Mahler measure (involving products of log |P| for possibly different P’s), and higher Mahler measure (involving logk |P|).

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2

Sasaki, Yoshitaka. "Zeta Mahler measures, multiple zeta values and L-values." International Journal of Number Theory 11, no. 07 (October 21, 2015): 2239–46. http://dx.doi.org/10.1142/s1793042115501006.

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The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

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3

Kurokawa, Nobushige. "A $q$-Mahler measure." Proceedings of the Japan Academy, Series A, Mathematical Sciences 80, no. 5 (May 2004): 70–73. http://dx.doi.org/10.3792/pjaa.80.70.

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4

Everest, G. R., and Bríd Ní Fhlathúin. "The elliptic Mahler measure." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 1 (July 1996): 13–25. http://dx.doi.org/10.1017/s0305004100074624.

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In this paper, we are going to introduce an elliptic analogue of the classical Mahler measure of an integral polynomial. The measure is shown to vanish if and only if the roots of the polynomial are attached to division points of the curve. This is the exact analogue of the statement that the Mahler measure of an integral polynomial vanishes if and only if all of its roots are division points of the circle (in other words, roots of unity). The proof of this result exploits an integral representation for the local canonical heights on the elliptic curve. The integral is that of a simple polynomial function over the complete curve in the appropriate valuation. Attention then shifts to the calculation of local integrals of arbitrary rational functions on elliptic curves. Results are proved which show these integrals may be computed as effective limits of Riemann sums. Finally, consideration is given to analogous behaviour for abelian varieties. We give a method, in principle, for computing the global canonical height of a rational point on an abelian variety denned over an algebraic number field, once again exploiting the integral representation of the local heights. The methods used include recent inequalities for linear forms in elliptic and abelian logarithms.

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5

Fei, Jiarui. "Mahler measure of 3D Landau–Ginzburg potentials." Forum Mathematicum 33, no. 5 (July 28, 2021): 1369–401. http://dx.doi.org/10.1515/forum-2020-0339.

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Abstract We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K ⁢ 3 {K3} hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families.

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6

Guilloux, Antonin, and Julien Marché. "Volume function and Mahler measure of exact polynomials." Compositio Mathematica 157, no. 4 (April 2021): 809–34. http://dx.doi.org/10.1112/s0010437x21007016.

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We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.

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7

SILVER, DANIEL S., and SUSAN G. WILLIAMS. "MAHLER MEASURE OF ALEXANDER POLYNOMIALS." Journal of the London Mathematical Society 69, no. 03 (May 24, 2004): 767–82. http://dx.doi.org/10.1112/s0024610704005289.

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8

PINNER, CHRISTOPHER. "Bounding the elliptic Mahler measure." Mathematical Proceedings of the Cambridge Philosophical Society 124, no. 3 (November 1998): 521–29. http://dx.doi.org/10.1017/s0305004198002795.

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9

Mossinghoff, Michael J. "Polynomials with small Mahler measure." Mathematics of Computation 67, no. 224 (October 1, 1998): 1697–706. http://dx.doi.org/10.1090/s0025-5718-98-01006-0.

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10

Amoroso, Francesco. "Mahler measure on Galois extensions." International Journal of Number Theory 14, no. 06 (July 2018): 1605–17. http://dx.doi.org/10.1142/s1793042118500963.

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We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

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11

Dubickas, A., and C. J. Smyth. "On the Metric Mahler Measure." Journal of Number Theory 86, no. 2 (February 2001): 368–87. http://dx.doi.org/10.1006/jnth.2000.2579.

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12

Rodriguez-Villegas, Fernando, Ricardo Toledano, and Jeffrey D. Vaaler. "ESTIMATES FOR MAHLER’S MEASURE OF A LINEAR FORM." Proceedings of the Edinburgh Mathematical Society 47, no. 2 (June 2004): 473–94. http://dx.doi.org/10.1017/s0013091503000701.

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AbstractLet $L_{\bm{a}}(\bm{z})=a_1z_1+a_2z_2+\cdots+a_Nz_N$ be a linear form in $N$ complex variables $z_1,z_2,\dots,z_N$ with non-zero coefficients. We establish several estimates for the logarithmic Mahler measure of $L_{\bm{a}}$. In general, we show that the logarithmic Mahler measure of $L_{\bm{a}}(\bm{z})$ and the logarithm of the norm of $\bm{a}$ differ by a bounded amount that is independent of $N$. We prove a further estimate which is useful for making an approximate numerical evaluation of the logarithmic Mahler measure.AMS 2000 Mathematics subject classification: Primary 11C08; 11Y35; 26D15

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13

Samuels, Charles L. "The Infimum in the Metric Mahler Measure." Canadian Mathematical Bulletin 54, no. 4 (December 1, 2011): 739–47. http://dx.doi.org/10.4153/cmb-2011-028-x.

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AbstractDubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number α by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.

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14

FLAMMANG, V. "THE MAHLER MEASURE OF TRINOMIALS OF HEIGHT 1." Journal of the Australian Mathematical Society 96, no. 2 (December 5, 2013): 231–43. http://dx.doi.org/10.1017/s1446788713000633.

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AbstractWe study the Mahler measure of the trinomials ${z}^{n} \pm {z}^{k} \pm 1$. We give two criteria to identify those whose Mahler measure is less than $1. 381\hspace{0.167em} 356\cdots = M(1+ {z}_{1} + {z}_{2} )$. We prove that these criteria are true for $n$ sufficiently large.

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15

Lalín, Matilde, and Gang Wu. "Regulator proofs for Boyd’s identities on genus 2 curves." International Journal of Number Theory 15, no. 05 (May 28, 2019): 945–67. http://dx.doi.org/10.1142/s1793042119500519.

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We use the elliptic regulator to recover some identities between Mahler measures involving certain families of genus 2 curves that were conjectured by Boyd and proven by Bertin and Zudilin by differentiating the Mahler measures and using hypergeometric identities. Since our proofs involve the regulator, they yield light into the expected relation of each Mahler measure to special values of [Formula: see text]-functions of certain elliptic curves.

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16

GON, YASURO, and HIDEO OYANAGI. "GENERALIZED MAHLER MEASURES AND MULTIPLE SINE FUNCTIONS." International Journal of Mathematics 15, no. 05 (July 2004): 425–42. http://dx.doi.org/10.1142/s0129167x04002363.

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We introduce a generalized Mahler measure. It has relations to multiple sine functions and Dirichlet L-functions. In particular, we are able to express special values of Dirichlet L-functions by sum of logarithmic generalized Mahler measures.

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17

Lalín, Matilde N., and Frank Ramamonjisoa. "The Mahler measure of a Weierstrass form." International Journal of Number Theory 13, no. 08 (August 2, 2017): 2195–214. http://dx.doi.org/10.1142/s1793042117501196.

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We prove an identity between Mahler measures of polynomials that was originally conjectured by Boyd. The combination of this identity with a result of Zudilin leads to a formula involving a Mahler measure of a Weierstrass form of conductor 17 given in terms of [Formula: see text]. Our proof involves a non-trivial identity between regulators which leads to the elliptic curve [Formula: see text]-function being expressed in terms of the regulator evaluated in a non-rational non-torsion point.

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18

Le, Thang T. Q. "Homology torsion growth and Mahler measure." Commentarii Mathematici Helvetici 89, no. 3 (2014): 719–57. http://dx.doi.org/10.4171/cmh/332.

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19

Benferhat, Leila. "Mahler Measure and Modular Elliptic Surfaces." Quaestiones Mathematicae 33, no. 2 (July 7, 2010): 231–43. http://dx.doi.org/10.2989/16073606.2010.491190.

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20

Garza, John. "The Mahler measure of dihedral extensions." Acta Arithmetica 131, no. 3 (2008): 201–15. http://dx.doi.org/10.4064/aa131-3-1.

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21

Lalín, Matilde N. "Equations for Mahler measure and isogenies." Journal de Théorie des Nombres de Bordeaux 25, no. 2 (2013): 387–99. http://dx.doi.org/10.5802/jtnb.841.

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22

Silver, Daniel S., Alexander Stoimenow, and Susan G. Williams. "Euclidean Mahler measure and twisted links." Algebraic & Geometric Topology 6, no. 2 (April 7, 2006): 581–602. http://dx.doi.org/10.2140/agt.2006.6.581.

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23

Everest, Graham, and Chris Pinner. "Bounding the Elliptic Mahler Measure II." Journal of the London Mathematical Society 58, no. 1 (August 1998): 1–8. http://dx.doi.org/10.1112/s0024610798006577.

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24

Biswas, Arunabha. "Asymptotic nature of higher Mahler measure." Acta Arithmetica 166, no. 1 (2014): 15–21. http://dx.doi.org/10.4064/aa166-1-2.

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25

SAUNDERS, J. C. "MAHLER MEASURE OF ‘ALMOST’ RECIPROCAL POLYNOMIALS." Bulletin of the Australian Mathematical Society 98, no. 1 (May 3, 2018): 70–76. http://dx.doi.org/10.1017/s0004972718000217.

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We give a lower bound of the Mahler measure on a set of polynomials that are ‘almost’ reciprocal. Here ‘almost’ reciprocal means that the outermost coefficients of each polynomial mirror each other in proportion, while this pattern may break down for the innermost coefficients.

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26

Silver, Daniel S., and Susan G. Williams. "Mahler measure, links and homology growth." Topology 41, no. 5 (September 2002): 979–91. http://dx.doi.org/10.1016/s0040-9383(01)00014-3.

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27

Vandervelde, Sam. "The Mahler measure of parametrizable polynomials." Journal of Number Theory 128, no. 8 (August 2008): 2231–50. http://dx.doi.org/10.1016/j.jnt.2007.12.002.

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28

Biswas, Arunabha, and Chris Monico. "Limiting value of higher Mahler measure." Journal of Number Theory 143 (October 2014): 357–62. http://dx.doi.org/10.1016/j.jnt.2014.04.015.

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29

Guillera, Jesús, and Mathew Rogers. "Mahler measure and the WZ algorithm." Proceedings of the American Mathematical Society 143, no. 7 (March 18, 2015): 2873–86. http://dx.doi.org/10.1090/s0002-9939-2015-12240-x.

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30

Lalín, Matilde N. "An algebraic integration for Mahler measure." Duke Mathematical Journal 138, no. 3 (June 2007): 391–422. http://dx.doi.org/10.1215/s0012-7094-07-13832-8.

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31

Dobrowolski, Edward, and Chris Smyth. "Mahler measures of polynomials that are sums of a bounded number of monomials." International Journal of Number Theory 13, no. 06 (December 5, 2016): 1603–10. http://dx.doi.org/10.1142/s1793042117500907.

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We study Laurent polynomials in any number of variables that are sums of at most [Formula: see text] monomials. We first show that the Mahler measure of such a polynomial is at least [Formula: see text], where [Formula: see text] is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with [Formula: see text] being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure [Formula: see text] is determined by its [Formula: see text] coefficients.

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32

Rogers, Mathew, and Wadim Zudilin. "FromL-series of elliptic curves to Mahler measures." Compositio Mathematica 148, no. 2 (January 23, 2012): 385–414. http://dx.doi.org/10.1112/s0010437x11007342.

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AbstractWe prove the conjectural relations between Mahler measures andL-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions forL-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y)(X+Y)−αXY,α∈ℝ.

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33

Akhtari, Shabnam, and Jeffrey D. Vaaler. "Lower bounds for Mahler measure that depend on the number of monomials." International Journal of Number Theory 15, no. 07 (July 21, 2019): 1425–36. http://dx.doi.org/10.1142/s1793042119500805.

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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 [Formula: see text] variables, our result depends on [Formula: see text] as an ordered group, and in general, our lower bound depends on the choice of ordering.

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34

Amoroso, Francesco. "On the Mahler measure in several variables." Bulletin of the London Mathematical Society 40, no. 4 (May 21, 2008): 619–30. http://dx.doi.org/10.1112/blms/bdn041.

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35

Champanerkar, Abhijit, Ilya Kofman, and Matilde Lalín. "Mahler measure and the Vol‐Det Conjecture." Journal of the London Mathematical Society 99, no. 3 (November 29, 2018): 872–900. http://dx.doi.org/10.1112/jlms.12200.

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36

Everest, Graham, and Chris Pinner. "Corrigendum: Bounding the Elliptic Mahler Measure II." Journal of the London Mathematical Society 62, no. 2 (October 2000): 640. http://dx.doi.org/10.1112/s0024610700001228.

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37

UEKI, JUN. "-adic Mahler measure and -covers of links." Ergodic Theory and Dynamical Systems 40, no. 1 (June 29, 2018): 272–88. http://dx.doi.org/10.1017/etds.2018.35.

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Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$-covers of rational homology 3-spheres branched over links. We obtain a $p$-adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coefficient of the Alexander polynomial, the $p$-adic entropy and the Iwasawa $\unicode[STIX]{x1D707}_{p}$-invariant. We also apply the purely $p$-adic theory of Besser–Deninger to $\mathbb{Z}$-covers of links. In addition, we study the entropies of profinite cyclic covers of links. We examine various examples throughout the paper.

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38

Fili, Paul, and Charles L. Samuels. "On the non-Archimedean metric Mahler measure." Journal of Number Theory 129, no. 7 (July 2009): 1698–708. http://dx.doi.org/10.1016/j.jnt.2008.12.009.

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39

Choi, Stephen, and Tamás Erdélyi. "Sums of monomials with large Mahler measure." Journal of Approximation Theory 197 (September 2015): 49–61. http://dx.doi.org/10.1016/j.jat.2014.01.003.

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40

Lal�n, Matilde. "Mahler Measure and Volumes in Hyperbolic Space." Geometriae Dedicata 107, no. 1 (August 2004): 211–34. http://dx.doi.org/10.1007/s10711-004-8123-8.

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41

Bertin, Marie José, and Wadim Zudilin. "On the Mahler measure of hyperelliptic families." Annales mathématiques du Québec 41, no. 1 (July 14, 2016): 199–211. http://dx.doi.org/10.1007/s40316-016-0068-4.

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42

Cochrane, Todd, R. M. S. Dissanayake, Nicholas Donohoue, M. I. M. Ishak, Vincent Pigno, Chris Pinner, and Craig Spencer. "Minimal Mahler Measure in Real Quadratic Fields." Experimental Mathematics 25, no. 2 (December 2, 2015): 107–15. http://dx.doi.org/10.1080/10586458.2015.1042599.

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43

Stulov, Konstantin, and Rongwei Yang. "An elementary inequality about the Mahler measure." Involve, a Journal of Mathematics 6, no. 4 (October 8, 2013): 393–97. http://dx.doi.org/10.2140/involve.2013.6.393.

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44

Choi, Kwok-Kwong Stephen, and Charles L. Samuels. "Two inequalities on the areal Mahler measure." Illinois Journal of Mathematics 56, no. 3 (2012): 825–34. http://dx.doi.org/10.1215/ijm/1391178550.

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45

Samart, Detchat. "Mahler Measures as Linear Combinations of L–values of Multiple Modular Forms." Canadian Journal of Mathematics 67, no. 2 (April 2015): 424–49. http://dx.doi.org/10.4153/cjm-2014-012-8.

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AbstractWe study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves L–values of at most one newform and/or at most one quadratic character. In this paper we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to L–values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular L–values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of L–functions.

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46

Dubickas, Artūras. "Mahler Measures Close to an Integer." Canadian Mathematical Bulletin 45, no. 2 (June 1, 2002): 196–203. http://dx.doi.org/10.4153/cmb-2002-022-8.

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AbstractWe prove that theMahler measure of an algebraic number cannot be too close to an integer, unless we have equality. The examples of certain Pisot numbers show that the respective inequality is sharp up to a constant. All cases when the measure is equal to the integer are described in terms of the minimal polynomials.

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47

Memić, Nacima. "Mahler coefficients of 1-Lipschitz measure-preserving functions on ℤp." International Journal of Number Theory 16, no. 06 (February 19, 2020): 1247–61. http://dx.doi.org/10.1142/s1793042120500645.

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In this work, we provide a complete description of Mahler coefficients of [Formula: see text]-Lipschitz measure-preserving functions on the ring of [Formula: see text]-adic integers [Formula: see text]. Our techniques are mainly based on some congruence identities including binomial coefficients. The main result provides an answer to one of Jeong’s conjectures, concerning a characterization of [Formula: see text]-Lipschitz measure-preserving functions by means of their Mahler coefficients. We provide an example showing that the formula mentioned in Jeong’s conjecture is sufficient but not necessary. Namely, we prove that there exist [Formula: see text]-Lipschitz measure-preserving functions that do not satisfy Jeong’s formula.

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48

Everest, G. R. "Estimating Mahler's measure." Bulletin of the Australian Mathematical Society 51, no. 1 (February 1995): 145–51. http://dx.doi.org/10.1017/s0004972700013976.

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In 1962, Mahler defined a measure for integer polynomials in several variables as the logarithmic integral over the torus. Many results exist about the values taken by the measure but many unsolved problems remain. In one variable, it is possible to express the measure as an effective limit of Riemann sums. We show that the same is true in several variables, using a non-obvious parametrisation of the torus together with Baker's Theorem on linear forms in logarithms of algebraic numbers.

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49

Flammang, V. "On the Zhang–Zagier measure." International Journal of Number Theory 14, no. 10 (October 25, 2018): 2663–71. http://dx.doi.org/10.1142/s1793042118501609.

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We first improve the known lower bound for the absolute Zhang–Zagier measure in the general case. Then we restrict our study to totally positive algebraic integers. In this case, we are able to find six points for the related spectrum. At last, we give inequalities involving the Zhang–Zagier measure, the Mahler measure and the length of such integers.

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50

BADZIAHIN, DZMITRY, and EVGENIY ZORIN. "On the irrationality measure of the Thue–Morse constant." Mathematical Proceedings of the Cambridge Philosophical Society 168, no. 3 (March 27, 2018): 455–72. http://dx.doi.org/10.1017/s0305004118000117.

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AbstractWe provide a non-trivial measure of irrationality for a class of Mahler numbers defined by infinite products. This class includes the Thue–Morse constant. Among other things, our results imply a generalisation to [12].

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