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Dissertations / Theses on the topic 'Mahler measure'
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1
Rogers, Mathew D. "Hypergeometric functions and Mahler measure." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/1420.
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The logarithmic Mahler measure of an n-variable Laurent
polynomial, P(x1,...,xn) is defined by [expression].
Using experimental methods, David Boyd conjectured a large number of
explicit relations between Mahler measures of polynomials and
special values of different types of L-series. This thesis
contains four papers which either prove or attempt to prove
conjectures due to Boyd. The introductory chapter contains an
overview of the contents of each manuscript.
2
Staines, Matthew. "On the inverse problem for Mahler Measure." Thesis, University of East Anglia, 2012. https://ueaeprints.uea.ac.uk/48118/.
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We investigate a number of aspects of the inverse problem for Mahler Measure. If β is an algebraic unit, we demonstrate how to determine if there are any reciprocal numbers with measure β. We also give a formula for the number of integer polynomials with measure β and given degree.
3
Chern, Shey-jey. "Estimates for the number of polynomials with bounded degree and bounded Mahler measure /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.
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De, Silva Dilum P. "Lind-Lehmer constant for groups of the form Z[superscript]n[subscript]p." Diss., Kansas State University, 2013. http://hdl.handle.net/2097/16244.
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Mohamed, Ismail Mohamed Ishak. "Lower bounds for heights in cyclotomic extensions and related problems." Diss., Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/2274.
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Mehrabdollahei, Mahya. "La mesure de Mahler d’une famille de polynômes exacts." Thesis, Sorbonne université, 2022. https://accesdistant.sorbonne-universite.fr/login?url=https://theses-intra.sorbonne-universite.fr/2022SORUS170.pdf.
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Dans cette thèse, nous étudions la suite de mesures de Mahler d’une famille de polynômes à deux variables exacts et réguliers, que nous notons Pd := P0≤i+j≤d xiyj . Elle n’est bornée ni en volume, ni en genre de la courbe algébrique sous-jacente. Nous obtenons une expression pour la mesure de Mahler de Pd comme somme finie de valeurs spéciales du dilogarithme de Bloch-Wigner. Nous utilisons SageMath pour approximer m(Pd) pour 1 ≤ d ≤ 1000. En recourant à trois méthodes différentes, nous prouvons que la limite de la suite de mesures de Mahler de cette famille converge vers 92π2 ζ(3). De plus, nous calculons le développement asymptotique de la mesure de Mahler de Pd et prouvons que sa vitesse de convergence est de O(log dd2 ). Nous démontrons également une généralisation du théorème de Boyd-Lawton, affirmant que les mesures de Mahler multivariées peuvent être approximéess en utilisant les mesures de Mahler de dimension inférieure. Enfin, nous prouvons que la mesure de Mahler de Pd pour d arbitraire peut être écrite comme une combinaison linéaire de fonctions L associées à un caractère de Dirichlet primitif impair. Nous calculons finalement explicitement la représentation de la mesure de Mahler de Pd en termes de fonctions L, pour 1 ≤ d ≤ 6<br>In this thesis we investigate the sequence of Mahler measures of a family of bivariate regular exact polynomials, called Pd := P0≤i+j≤d xiyj , unbounded in both degree and the genus of the algebraic curve. We obtain a closed formula for the Mahler measure of Pd in termsof special values of the Bloch–Wigner dilogarithm. We approximate m(Pd), for 1 ≤ d ≤ 1000,with arbitrary precision using SageMath. Using 3 different methods we prove that the limitof the sequence of the Mahler measure of this family converges to 92π2 ζ(3). Moreover, we compute the asymptotic expansion of the Mahler measure of Pd which implies that the rate of the convergence is O(log dd2 ). We also prove a generalization of the theorem of the Boyd-Lawton which asserts that the multivariate Mahler measures can be approximated using the lower dimensional Mahler measures. Finally, we prove that the Mahler measure of Pd, for arbitrary d can be written as a linear combination of L-functions associated with an odd primitive Dirichlet character. In addition, we compute explicitly the representation of the Mahler measure of Pd in terms of L-functions, for 1 ≤ d ≤ 6
7
Santos, Jefferson Marques. "Altura e equidistribuição de pontos algébricos." Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/7564.
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Previous issue date: 2017-06-20<br>The concept of roots of a polynomial is quite simple but has several applications. This concept extends more generally to the case of "small" algebraic points sequences in a curve. This dissertation aims to estimate the size of algebraic numbers by means of Weil height. In addition to showing that they are distributed evenly around the unit circle, through Bilu Equidistribution Theorem.<br>O conceito de raízes de um polinômio é bastante simples mas possui várias aplicações. Este conceito se estende de forma mais geral para o caso de sequências de pontos algébricos “pequenos” em uma curva. Esta dissertação tem por objetivo estimar o tamanho de números algébricos por meio da altura de Weil. Além de mostrar que os mesmos se distribuem uniformemente em torno do círculo unitário, por meio do Teorema de Equidistribuição de Bilu.
8
Fhlathuin, Brid ni. "Mahler's measure on Abelian varieties." Thesis, University of East Anglia, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.296951.
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This thesis is a study of the integration of proximity functions over certain compact groups. Mean values are found of the ultrametric valuation of certain rational functions associated with a divisor on an abelian variety, and it is shown how these may be expressed in terms of an integral, thus finding the analogue, for an abelian variety, of Mahler's definition of the measure of a polynomial. These integrals are shown to arise in a manner which mimics classical Riemann sums, and their relation with the global canonical height is investigated. It is shown that the measure is a rational multiple of log p. Similar results are given for elliptic curves, taking the divisor to be the identity of the group law, and somewhat stronger mean value theorems proven in this more specific case by working directly with local canonical heights rather than approaching them through related functions. Effective asymptotic formulae for the local height are derived, first for the kernel of reduction of a curve and then, via a detailed analysis of the local reduction of the curve, for the group of rational points. The theory of uniform distribution is used to show that the mean value also takes an integral form in the case of an archimedean valuations, and recent inequalities for elliptic forms in logarithms are used to give error terms for the convergence towards the measure. This is undertaken first for the local height on an elliptic curve, and then, in terms of general theta-functions, on an abelian variety. We then seek to exploit these generalisations of the Mahler measure to yield an alternative method to that of Silverman and Tate for the determining of the global height. The integration over a cyclic group of the laws satisfied locally by the height allows us to reformulate our theorems in a manner conducive to practical application. It is demonstrated how our asymptotic formulae may be used together with an appropriate computer software package, PARI in our case, to calculate the mean value of heights, and, more generally, of rational functions, on an elliptic curve and on abehan varieties of higher genus. Some such calculations are displayed, with comments on their efficacy and their possible future development.
9
Condon, John Donald. "Mahler measure evaluations in terms of polylogarithms." Thesis, 2004. http://hdl.handle.net/2152/1218.
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Condon, John Donald Rodríguez Villegas Fernando. "Mahler measure evaluations in terms of polylogarithms." 2004. http://wwwlib.umi.com/cr/utexas/fullcit?p3142710.
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Roy, Subham. "Generalized Mahler measure of a family of polynomials." Thèse, 2019. http://hdl.handle.net/1866/23797.
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Le présent mémoire traite une variation de la mesure de Mahler où l'intégrale de définition est réalisée sur un tore plus général. Notre travail est basé sur une famille de polynômes tempérée originellement étudiée par Boyd, P_k (x, y) = x + 1/x + y + 1/y + k avec k ∈ R_{>4}. Pour le k = 4 cas, nous utilisons des valeurs spéciales du dilogarithme de Bloch-Wigner pour obtenir la mesure de Mahler de P_4 sur un tore arbitraire (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } avec a, b ∈ R_{> 0}. Ensuite, nous établissons une relation entre la mesure de Mahler de P_8 sur un tore (T_ {a, √a} )^2 et sa mesure de Mahler standard. La combinaison de cette relation avec des résultats de Lalin, Rogers et Zudilin conduit à une formule impliquant les mesures de Mahler généralisées de ce polynôme données en termes de L' (E, 0). Au final, nous proposons une stratégie pour prouver des résultats similaires dans le cas général k> 4 sur (T_ {a, b})^2 avec certaines restrictions sur a, b.<br>In this thesis we consider a variation of the Mahler measure where the defining integral is performed over a more general torus. Our work is based on a tempered family of polynomials originally studied by Boyd, Boyd P_k (x, y) = x + 1/x + y + 1/y + k with k ∈ R_{>4}. For the k = 4 case we use special values of the Bloch-Wigner dilogarithm to obtain the Mahler measure of P_4 over an arbitrary torus (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } with a, b ∈ R_{> 0}. Next we establish a relation between the Mahler measure of P_8 over a torus(T_ {a, √a} )^2 and its standard Mahler measure. The combination of this relation with results due to Lalin, Rogers, and Zudilin leads to a formula involving the generalized Mahler measure of this polynomial given in terms of L'(E, 0). In the end, we propose a strategy to prove some similar results for the general case k > 4 over (T_ {a, b})^2 with some restrictions on a, b.
12
Gu, Jarry. "Polylogarithmes et mesure de Mahler." Thesis, 2020. http://hdl.handle.net/1866/24344.
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Le but principal de ce mémoire est de calculer la mesure de Mahler logarithmique d’une famille de polynômes à trois variables x^n + 1 + (x^(n−1) + 1)y + (x − 1)z. Pour réaliser cet objectif, on intègre des régulateurs définis sur des complexes motiviques polylogarithmiques. Pour comprendre ces régulateurs, on explore les propriétés des polylogarithmes et démontre quelques identités polylogarithmiques. Ensuite, on utilise les régulateurs afin de simplifier l’intégrante. Notre résultat est une formule qui relie la mesure de Mahler de la famille de polynômes susmentionnée au dilogarithme de Bloch–Wigner et à la fonction zêta de Riemann.<br>The main purpose of this thesis is to compute the logarithmic Mahler measure of the
three variable polynomial family xn + 1 + (xn−1 + 1)y + (x − 1)z. In order to accomplish
this, we integrate regulators defined on polylogarithmic motivic complexes. To understand
these regulators, we explore the properties of polylogarithms and show some polylogarithmic
identities. The regulators are then applied to simplify the integrand. Our result is a formula
relating the Mahler measure of the family of polynomials to the Bloch–Wigner Dilogarithm
and the Riemann zeta function.
13
Lalín, Matilde Noemí Rodriguez-Villegas Fernando. "Some relations of Mahler measure with hyperbolic volumes and special values of L-functions." 2005. http://repositories.lib.utexas.edu/bitstream/handle/2152/1971/lalinm19850.pdf.
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Lalín, Matilde Noemí. "Some relations of Mahler measure with hyperbolic volumes and special values of L-functions." Thesis, 2005. http://hdl.handle.net/2152/1971.
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Miner, Zachary Layne. "Norms extremal with respect to the Mahler measure and a generalization of Dirichlet's unit theorem." Thesis, 2011. http://hdl.handle.net/2152/ETD-UT-2011-05-3197.
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In this thesis, we introduce and study several norms constructed to satisfy an extremal property with respect to the Mahler measure. These norms naturally generalize the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer's conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures. Next, we generalize Dirichlet's S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a Q-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over Q retain their linear independence over R.<br>text
16
Lechasseur, Jean-Sébastien. "Mesure de Mahler supérieure de certaines fonctions rationelles." Thèse, 2012. http://hdl.handle.net/1866/8989.
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Nous exprimons la mesure de Mahler 2-supérieure et 3-supérieure de certaines fonctions rationnelles en terme de valeurs spéciales de la fonction zêta, de fonctions L et de polylogarithmes multiples. Les résultats obtenus sont une généralisation de ceux obtenus dans [10] pour la mesure de Mahler classique.
On améliore un de ces résultats en réduisant une combinaison linéaire de polylogarithmes
multiples en termes de valeurs spéciales de fonctions L. On termine avec la
réduction complète d’un cas particuler.<br>The 2-higher and 3-higher Mahler measure of some rational functions are given in terms
of special values of the Riemann zeta function, a Dirichlet L-function and multiple polylogarithms. Our results generalize those obtained in [10] for the classical Mahler measure.
We improve one of our results by providing a reduction for a certain linear combination
of multiple polylogarithms in terms of Dirichlet L-functions. We conclude by
giving a complete reduction of a special case.
17
Fili, Paul Arthur. "Orthogonal decompositions of the space of algebraic numbers modulo torsion." Thesis, 2010. http://hdl.handle.net/2152/ETD-UT-2010-05-1416.
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We introduce decompositions determined by Galois field and degree of the space of algebraic numbers modulo torsion and the space of algebraic points on an elliptic curve over a number field. These decompositions are orthogonal with respect to the natural inner product associated to the L² Weil height recently introduced by Allcock and Vaaler in the case of algebraic numbers and the inner product naturally associated to the Néron-Tate canonical height on an elliptic curve. Using these decompositions, we then introduce vector space norms associated to the Mahler measure. For algebraic numbers, we formulate L[superscript p] Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p=1 case and the Schinzel-Zassenhaus conjecture in the p=[infinity] case.<br>text
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Giard, Antoine. "La mesure de Mahler d’une forme de Weierstrass." Thèse, 2019. http://hdl.handle.net/1866/22549.
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Issa, Zahraa. "A Generalization of a Theorem of Boyd and Lawton." Thèse, 2012. http://hdl.handle.net/1866/8745.
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Ce mémoire s’applique à étudier d’abord, dans la première partie, la mesure de Mahler des polynômes à une seule variable. Il commence en donnant des définitions et quelques résultats pertinents pour le calcul de telle hauteur.
Il aborde aussi le sujet de la question de Lehmer, la conjecture la plus célèbre dans le domaine, donne quelques exemples et résultats ayant pour but de résoudre la question.
Ensuite, il y a l’extension de la mesure de Mahler sur les polynômes à plusieurs variables, une démarche semblable au premier cas de la mesure de Mahler, et le sujet des points limites avec quelques exemples.
Dans la seconde partie, on commence par donner des définitions concernant un ordre supérieur de la mesure de Mahler, et des généralisations en passant des polynômes simples aux polynômes à plusieurs variables.
La question de Lehmer existe aussi dans le domaine de la mesure de Mahler supérieure, mais avec des réponses totalement différentes.
À la fin, on arrive à notre objectif, qui sera la démonstration de la généralisation d’un théorème de Boyd-Lawton, ce dernier met en évidence une relation entre la mesure de Mahler des polynômes à plusieurs variables avec la limite de la mesure de Mahler des polynômes à une seule variable.
Ce résultat a des conséquences en termes de la conjecture de Lehmer et sert à clarifier la relation entre les valeurs de la mesure de Mahler des polynômes à une variable et celles des polynômes à plusieurs variables, qui, en effet, sont très différentes en nature.<br>This thesis applies to study first, in part 1, the Mahler measure of polynomials in one variable. It starts by giving some definitions and results that are important for calculating this height.
It also addresses the topic of Lehmer’s question, an interesting conjecture in the field, and it gives some examples and results aimed at resolving the issue.
The extension of the Mahler measure to several variable polynomials is then considered including the subject of limit points with some examples.
In the second part, we first give definitions of a higher order for the Mahler measure, and generalize from single variable polynomials to multivariable polynomials.
Lehmer’s question has a counterpart in the area of the higher Mahler measure, but with totally different answers.
At the end, we reach our goal, where we will demonstrate the generalization of a theorem of Boyd-Lawton. This theorem shows a relation between the limit of Mahler measure of multivariable polynomials with Mahler measure of polynomials in one variable.
This result has implications in terms of Lehmer's conjecture and serves to clarify the relationship between the Mahler measure of one variable polynomials, and the Mahler measure of multivariable polynomials, which are very different.