Integrating onp-adic Lie groups

Inspired by the classical Mahler measure of a polynomial, we study the integral of the order of an arithmetic polynomial on a compactp-adic Lie group. A result of Denef and van den Dries guarantees this is always a rational number. Integrals of this kind arise naturally; for example, the local canonical height of a rational point on an elliptic curve is given by a Mahler measure. Also, the mean valuation of the normal integral generators in a finite Galois extension arises as a Mahler measure. There is interest in being able to calculate the value of this measure. We show that for some classical groups, it is possible to reduce the integral to a simpler form, one where explicit computations are feasible. The motivation comes from the calculus trick of integration by substitution, also from Weyl’s criterion. Applications are given to Galois Module Theory. Also, a close encounter with Leopoldt’s conjecture is recorded. We deduce our results on the Mahler measure from the more general setting of local zeta functions defined forp-adic Lie groups. Our techniques apply to certain zeta functions, so we state and prove our results at that level of generality in our main theorem. Thanks go to Steve Wilson, the SERC and the London Mathematical Society for the Durham Galois Modules Workshop, which inspired the results in §5. Thanks go to Alex Lubotzky and the Royal Society for making possible the visit of the second author to the Hebrew University in Jerusalem which lead to the zeta-function point of view in §1 and §2.     

Mahler measures and computations with regulators

In this work we apply the techniques that were developed in [M.N. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), in press] in order to study several examples of multivariable polynomials whose Mahler measure is expressed in terms of special values of the Riemann zeta function or Dirichlet L-series. The examples may be understood in terms of evaluations of regulators. Moreover, we apply the same techniques to the computation of generalized Mahler measures, in the sense of Gon and Oyanagi [Y. Gon, H. Oyanagi, Generalized Mahler measures and multiple sine functions, Internat. J. Math. 15 (5) (2004) 425-442].     

The Mahler measure of a Calabi–Yau threefold and special $$$$-values

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi–Yau threefold. We show that its Mahler measure is a rational linear combination of a special \(L\)-value of the normalized newform in \(S_4(\Gamma _0(8))\) and a Riemann zeta value. This is equivalent to a new formula for a \(_6F_5\)-hypergeometric series evaluated at 1.     

On Numbers which are Mahler Measures

We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer.     

Mahler measures in a cubic field

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.     

A formula for the Mahler measure of axy+bx+cy+d

We introduce a formula for the Mahler measure of axy+bx+cy+d with complex coefficients a,b,c, and d and give examples which demonstrate a connection with L-functions. We then prove a generalization of Maillot's formula when the coefficients are real. Next, we discuss operations on the coefficients which fix the Mahler measure. Finally, we prove an alternate formulation of the main result in order to calculate the Mahler measure of a two-parameter family of polynomials in three variables.     

Three-variable Mahler measures and special values of L-functions of modular forms

The Ramanujan Journal - In this paper we study the Mahler measures of two families of Laurent polynomials. We prove several three-variable Mahler measure formulas initially conjectured by D. Samart.     

On the Mahler measure of resultants in small dimensions

We prove that sparse resultants having Mahler measure equal to zero are those whose Newton polytope has dimension one. We then compute the Mahler measure of resultants in dimension two, and examples in dimension three and four. Finally, we show that sparse resultants are tempered polynomials. This property suggests that their Mahler measure may lead to special values of -functions and polylogarithms.     

Norms extremal with respect to the Mahler measure

In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of 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.     

Some families of links with divergent Mahler measure

In the following note we develop a method to prove that the Mahler Measure of the Jones polynomial of a family of links diverges. We apply this to several examples from the literature. We then use the W-polynomial to find the Kauffman Bracket of some families of Montesinos links and show that their Jones polynomials too have divergent Mahler measure.     

Polynomials with small Mahler measure

We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than , test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near , four new Salem numbers less than , and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer's degree 10 polynomial.

     

Certain Diophantine Properties of the Mahler Measure

It is proved that a polynomial in several Mahler measures with positive rational coefficients is equal to an integer if and only if all these Mahler measures are integers. An estimate for the distance between a metric Mahler measure and an integer is obtained. Finally, it is proved that the ratio of two distinct Mahler measures of algebraic units is irrational.     

A combinatorial problem related to Mahler's measure

We give a generalization of a result of Myerson on the asymptoticbehavior of norms of certain Gaussian periods. The proof exploitsproperties of the Mahler measure of a trinomial.     

On a certain geometric mean of the values of a polynomial

The maximum of the geometric mean of the values of a polynomial in the vertices of a regularkgon inscribed into the unit circle is greater than or equal to its Mahler measure. It also tends to the Mahler measure ask tends to infinity. We give quantitative versions of this statement: the upper bounds for the ratio of these two quantities. Partially supported by the Lithuanian State Science and Studies Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 17–27, January–March, 2000.     

Weak log-majorization, Mahler measure and polynomial inequalities

We explore the use of the weak log-majorization order in the analytic theory of polynomials. We examine the relationship between weak log-majorization and Mahler measure. We also improve the weak log-majorization form of the de Bruijn-Springer-Mahler inequality.     

New directions in algebraic dynamical systems

The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.     

The Mahler Measure of the Rudin–Shapiro Polynomials

Littlewood polynomials are polynomials with each of their coefficients in \(\{-1,1\}\). A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin–Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin–Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin–Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin–Shapiro polynomials have been known before.     

Jones多项式的零点

利用数论理论证明了纽结的Jones多项式仅有可能的有理根是O,而链环的Jones多项式仅有可能的有理根是0和-1.给出了作为Jones多项式根的所有可能单位根,以及所有可能的具有平凡Mahler测度的Jones多项式.最后指出了交叉数不超过11的纽结中,只有4_1,8_9,9_(42),K11n19的Jones多项式具有平凡的Mahler测度,从而回答了林晓松提出的关于Mahler测度的一个问题.     

Lehmer's problem for compact Abelian groups

We formulate Lehmer's Problem concerning the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components, then its Lehmer constant vanishes.

     

Higher Mahler measure for cyclotomic polynomials and Lehmer’s question

The k-higher Mahler measure of a non-zero polynomial P is the integral of log  ^k |P| on the unit circle. In this note, we consider Lehmer’s question (which is a long-standing open problem for k=1) for k>1 and find some interesting formulas for 2- and 3-higher Mahler measure of cyclotomic polynomials.