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We study the Mahler measures of the polynomial family Q_k(x,y) = x^3+y^3+1-kxy using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers \leqslant 3, we employ these points to derive interesting formulas that link the Mahler measures of Q_k(x,y) to L-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure \tilde {n}(k) introduced by Samart recently. For k=\sqrt [3]{729\pm 405\sqrt {3}}, we also prove an equality that expresses a 2\times 2 determinant with entries the Mahler measures of Q_k(x,y) as some multiple of the L-value of two isogenous elliptic curves over \mathbb {Q}(\sqrt {3}). [ABSTRACT FROM AUTHOR]

Using the special value at $u=1$ of Artin–Ihara L -functions, we associate to every $\mathbb {Z}$ -cover of a finite connected graph a polynomial, which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce–Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specialising to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I -graphs (including the generalised Petersen graphs). We also express the p -adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p -adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb {Z}$ -cover, our result gives us back Lengyel's calculation of the p -adic valuations of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

Following the work of Lalín and Mittal on the Mahler measure over arbitrary tori, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in two variables when they do not vanish on the integration torus. We establish certain relations between the standard Mahler measure and the generalized Mahler measure of such polynomials. Later we focus our investigation on a tempered family of polynomials originally studied by Boyd, namely Q r (x , y) = x + 1 x + y + 1 y + r with r ∈ C , and apply our results to this family. For the r = 4 case, we explicitly calculate the generalized Mahler measure of Q 4 over any arbitrary torus in terms of special values of the Bloch–Wigner dilogarithm. Finally, we extend our results to the several variable setting. [ABSTRACT FROM AUTHOR]

For every algebraic number κ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of f κ (z) = log | z − κ | at the conjugates are essentially bounded from above by − h (κ). This completes a characterization on functions f κ initiated by Autissier and Baker–Masser, who cover the cases κ = 2 and | κ | ≠ 1 , respectively. Using the same ideas we also prove analogues in the p -adic setting. [ABSTRACT FROM AUTHOR]

In this note we study the Mahler measures of reciprocal polynomials P k = x + 1 / x + y + 1 / y + k for k = 4 ± 4 2 . We prove identities relating the Mahler measure of P 4 ± 4 2 to special values of the L -functions of weight 2 cusp forms of level 64. We also express the Beilinson regulator which is a 2 × 2 determinant of Mahler measures as special value of L -function of the elliptic curve defined by P 4 + 4 2 = 0. [ABSTRACT FROM AUTHOR]

We exhibit some nontrivial evaluations of the areal Mahler measure of multivariable polynomials, defined by Pritsker [Pri08] by considering the integral over the product of unit disks instead of the unit torus as in the standard case. As in the case of the classical Mahler measure, we find examples yielding special values of L -functions. [ABSTRACT FROM AUTHOR]

Let α be a non-zero algebraic unit which is not a root of unity and K be a number field of degree d over Q . In this paper, we prove the following: Let P be a prime ideal of O K which lies above a rational odd prime p such that P O K (α) = P 1 e 1 P 2 e 2 ⋯ P g e g , where max 1 ≤ i ≤ g { e i } ≤ p and e 1 + ⋯ + e g = [ K (α) : K ]. Then h (α) ≥ c , where c > 0 is an effectively computable constant depending only on p and [ K : Q ] = d. This generalizes a result of Petsche. Also, we prove the following: Let P be a prime ideal of O K which lies above 2 such that P O K (α) = P 1 e 1 P 2 e 2 ⋯ P r e r , where e 1 + ⋯ + e r = [ K (α) : K ]. Then M (α) ≥ C (K) , where C (K) > 1 is a constant which depends only on [ K : Q ] = d. [ABSTRACT FROM AUTHOR]

In this paper, we use Padé approximations, constructed in [Kawashima and Poëls, Padé approximations for shifted functions and parametric geometry of numbers, J. Number Theory 243 (2023) 646–687] for binomial functions, to give a new upper bound for the number of the solutions of the S -unit equation in two variables. Combining explicit Padé approximants with a simple argument relying on Mahler measure as well as the local height, we refine the bound due to Evertse [On equations in S -units and the Thue-Mahler equation, Invent. Math. 75 (1984) 561–584]. [ABSTRACT FROM AUTHOR]

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $ -function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$ , respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points. [ABSTRACT FROM AUTHOR]

In this note we consider various generalizations of the classical Mahler measure M(P)=\exp \left (\frac {1}{2\pi }\int _0^{2\pi } \log |P(e^{i\theta })\,d\theta \right) for complex polynomials P, and prove sharp lower bounds for them. For example, we show that for any monic polynomial P \in {\mathbb {C}}[z] satisfying |P(0)|=1 the quantity M_0(P) = \exp \left (\frac {1}{2\pi }\int _0^{2\pi } \max (0,\log |P(e^{i\theta })|)\,d\theta \right) is greater than or equal to M(1+z_1+z_2)=1.381356\dots. This inequality is best possible, with equality being attained for all P(z)=z^n+c with n \in \mathbb {N} and |c|=1. [ABSTRACT FROM AUTHOR]

Recently, the second author and Qin numerically verified some Mahler measure identities of genus 2 and 3 polynomial families. In this paper, we use the elliptic regulator to prove an identity involving shifted Mahler measure for Boyd's family. [ABSTRACT FROM AUTHOR]

Let α be a non-zero algebraic integer of degree d which is not a root of unity. We prove that, if there exists an odd prime p with either (1) p ≤ d + 1 and p O Q (α) = P 1 P 2 ⋯ P d where P 1 , ... , P d are distinct prime ideals of O Q (α) , or (2) p ≤ d and p O Q (α) = P 1 e 1 P 2 e 2 ⋯ P g e g , where max 1 ≤ i ≤ g { e i } ≤ p and ∑ i = 1 g e i = d , then M (α) ≥ p / 2. We also prove that if the residual degrees of primes in O Q (α) which are lying above 2 are 1 , then M (α) ≥ 2 1 / 4 . This generalizes a result of Garza. [ABSTRACT FROM AUTHOR]

We prove that for each odd integer d ≥ 3 there are infinitely many number fields K of degree d such that each generator α of K has Mahler measure greater than or equal to d-d|ΔK|d+1/d(2d-2), where ΔK is the discriminant of the field K. This, combined with an earlier result of Vaaler and Widmer for composite d, answers negatively a question of Ruppert raised in 1998 about 'small' algebraic generators for every d ≥ 3. We also show that for each d ≥ 2 and any ε > 0, there exist infinitely many number fields K of degree d such that every algebraic integer generator α of K has Mahler measure greater than (1 - ε|ΔK|1/d. On the other hand, every such field K contains an algebraic integer generator α with Mahler measure smaller that |ΔK|1/d. This generalizes the corresponding bounds recently established by Eldredge and Petersen for d = 3. [ABSTRACT FROM AUTHOR]

The minimal integral Mahler measure of a number field K , M ( K) , is the minimal Mahler measure of an integral generator of K. Upper and lower bounds, which depend on the discriminant and degree of K , are known. We show that for three natural families of cubics, the lower bounds are sharp with respect to its growth as a function of discriminant. We construct an algorithm to compute M ( K) for all cubics with absolute value of the discriminant bounded by N and show the resulting data for N = 1 0 , 0 0 0. [ABSTRACT FROM AUTHOR]

In this paper we consider the set of Mahler measures of generalized Pisot and Salem numbers and give a characterization of such numbers in terms of their minimal polynomials. We show that every real quadratic algebraic integer β can be expressed by the difference of the Mahler measure of a generalized Pisot number and a Pisot number in ℚ(β). [ABSTRACT FROM AUTHOR]

For an (irreducible) recurrence equation with coefficients from $\mathbb Z[n]$ and its two linearly independent rational solutions $u_n,v_n$ , the limit of $u_n/v_n$ as $n\to \infty $ , when it exists, is called the Apéry limit. We give a construction that realises certain quotients of L-values of elliptic curves as Apéry limits. [ABSTRACT FROM AUTHOR]

The k -generalized Fibonacci and Pell polynomials are the polynomials X k − X k − 1 − X k − 2 − ⋯ − 1 and X k − 2 X k − 1 − X k − 2 − ⋯ − 1 , respectively. Here, k ≥ 2 is any integer. In this paper, we show that any two roots of some generalized Fibonacci and Pell polynomials are multiplicatively independent confirming a conjecture from [Bravo, Herrera and Luca, Common values of generalized Fibonacci and Pell sequences, J. Number Theory  226 (2021) 51–71]. [ABSTRACT FROM AUTHOR]

Let α be a nonzero algebraic integer of degree d whose all conjugates α 1 = α , α 2 , ... , α d lie in a sector | arg ⁡ z | ≤ θ , 0 < θ < 90 °. In 2013, K. Stulov and R. Yang defined the total distance of α as td (α) = ∑ i = 1 d | | α i | − 1 |. This paper is organized in two parts. First, we compute the greatest lower bound c (θ) of the absolute total distance of α , for α belonging to twenty one subintervals of (0 , 90). Among these subintervals, eighteen are complete. Moreover, there are two sequences of seven then nine both complete and consecutive subintervals. It is the first time that these phenomena appear: such a large number of subintervals, such a large number of complete subintervals and such a large number of complete and consecutive subintervals. These phenomena are keeping with a conjecture of G. Rhin and C. J. Smyth on the nature of the function c (θ). Secondly, using the previous results and assuming that the above conjecture is true, we give upper bounds and lower bounds for the total distance involving the Mahler measure. Mostly these bounds improve those of K. Stulov and R. Yang. All computations are done by using the method of explicit auxiliary functions. The polynomials involved in these functions are found by our recursive algorithm. [ABSTRACT FROM AUTHOR]

Let \lambda \in (1,\sqrt {2}] be an algebraic integer with Mahler measure 2. A classical result of Garsia shows that the Bernoulli convolution \mu _\lambda is absolutely continuous with respect to the Lebesgue measure with a density function in L^\infty. In this paper, we show that the density function is continuous. [ABSTRACT FROM AUTHOR]

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks. Afterwards we deal with higher-dimensional quantum walks. For comparison with the case of the quantum walk, we also treat the case of higher-dimensional random walks. Our results bridge between the Mahler measure and the zeta function via quantum walks for the first time. [ABSTRACT FROM AUTHOR]

In this short note, we study Lehmer's conjecture in terms of stability theory. We state Bounded Lehmer's conjecture, and we prove that if a certain formula is uniformly stable in a class of structures, then Bounded Lehmer's conjecture holds. Our proof is based on Van der Waerden's theorem from additive combinatorics. [ABSTRACT FROM AUTHOR]

We study the zeta Mahler function (ZMF, also zeta Mahler measure), which is closely related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials k + (x 1 + x 1 - 1) ⋯ x r + x r - 1 , where k is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore connections with the Mahler measure. [ABSTRACT FROM AUTHOR]

Motivated by a famous question of Lehmer about the Mahler measure, we study and solve its analytic analogue. [ABSTRACT FROM AUTHOR]

Cogdell et al. ['Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space', Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , where $J_0$ is the order-zero Bessel function of the first kind and a and $r_m$ are positive real numbers. To facilitate our proof we develop an alternative description of the integral's asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix. [ABSTRACT FROM AUTHOR]

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of the Boyd–Lawton formula, and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true. [ABSTRACT FROM AUTHOR]

Let α be a Salem number with conjugates , where α1 := α. We prove some properties of the products of the form α1α2· · · αn, called, by A. Dubickas and C.J. Smyth, Salem half-norms. This allows us to complete, partially, two related results due to C. Christopoulos and J. McKee about the Galois group of the splitting field of the minimal polynomial of α, and to A. Dubickas on a question of D.W. Boyd about Salem numbers which are Mahler measures of non-reciprocal algebraic integers. [ABSTRACT FROM AUTHOR]

Inspired by Szegö’s theorem, this paper focuses on characterizing cyclic vectors and outer functions in H 2 (D 2) by means of some geometric quantities and Mahler measure for functions. Under some mild conditions, various intriguing descriptions are obtained to investigate function-theoretic properties of these quantities. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

I share some of my adventures in mathematical research and homeschooling in the time of COVID-19. [ABSTRACT FROM AUTHOR]

In this work, we are dealing with some properties relating the zeros of a polynomial and its Mahler measure. We provide estimates on the number of real zeros of a polynomial, lower bounds on the distance between the zeros of a polynomial and non-zeros located on the unit circle and a lower bound on the number of zeros of a polynomial in the disk { | x − 1 | < 1 }. [ABSTRACT FROM AUTHOR]

For each algebraic number α and each positive real number t , the t -metric Mahler measure m t (α) creates an extremal problem whose solution varies depending on the value of t. The second author studied the points t at which the solution changes, called exceptional points for α. Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every N ∈ ℕ , there exists a number having at least N exceptional points. In this paper, we describe a polynomial time algorithm for establishing the existence of numbers with at least N exceptional points. Our work constitutes an improvement over the best known existing algorithm which requires exponential time. We apply our main result to show that there exist numbers with at least 3 7 exceptional points, another improvement over previous work which was only able to reach 1 1 exceptional points. [ABSTRACT FROM AUTHOR]

We use the elliptic regulator to prove an identity between the Mahler measures of a genus 3 polynomial family and of a genus 1 polynomial family that was initially conjectured by Liu and Qin. [ABSTRACT FROM AUTHOR]

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k ∈ { 0 , d } or gcd (k , d) = 1 . We therefore study the behaviour in the case where 0 < k < d and gcd (k , d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove a formula relating the Mahler measure of an infinite family of three-variable polynomials to a combination of the Riemann zeta function at s = 3 and special values of the Bloch–Wigner dilogarithm by evaluating a regulator. The evaluation requires two different applications of Jensen's formula and analyzing the integral in two different planes, as opposed to the more common strategy of using only one plane. The degrees of the monomials involving one of the variables are allowed to vary freely, leading to an interesting application of the Boyd–Lawton formula. [ABSTRACT FROM AUTHOR]

Lehmer’s question, an open question about the Mahler measure of monic integral polynomials, is shown to be equivalent to a question about the complexity growth rate of signed 1-periodic graphs. If G is a d-periodic graph (i.e. G has a co-finite free ℤd-action by automorphisms), then a d-variable polynomial ∆G can be defined with Mahler measure equal to the logarithmic growth rate γG of a complexity defined for the finite quotients of G. A plane 1-periodic graph determines a link via projection and the medial graph construction. The polynomial ∆G can be determined from the Alexander polynomial of the link. The complexity growth rate γG of any d-periodic graph is at least log 2. An investigation of plane 1- and 2-periodic graphs yields more connections with knot theory including work of A. Champanerkar and I. Kofman. [ABSTRACT FROM AUTHOR]

We establish a functional identity for Mahler measures of the two-parametric family P a , c (x , y) = a (x + 1 / x) + y + 1 / y + c. Our result extends an identity proven in a paper of Lalín, Zudilin and Samart. As a by-product, we obtain evaluations of m (P a , c) for some algebraic values of a and c in terms of special values of L-functions and logarithms. We also give a sufficient condition for validity of a certain identity between the elliptic integrals of the first and the third kind, which implies several identities for m (P a , c). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For an algebraic number α we denote by M (α) the Mahler measure of α . As M (α) is again an algebraic number (indeed, an algebraic integer), M (·) is a self-map on Q ¯ , and therefore defines a dynamical system. The orbit size of α , denoted # O M (α) , is the cardinality of the forward orbit of α under M. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size. [ABSTRACT FROM AUTHOR]

In this paper we give an estimate for the difference between the moduli of two roots of a polynomial with integer coefficients in terms of its degree and Mahler measure. An application of this estimate implies a stronger version of a recent result of Gómez Ruiz and Luca. In passing, we prove an estimate for the distance between two algebraic numbers in terms of their degrees and Mahler measures. For possible applications all our results are given with explicit constants. They are stated without any extra conditions or unnecessary assumptions whenever possible. [ABSTRACT FROM AUTHOR]

We prove an identity between two Mahler measures. Combining it with a result of Rogers and Zudilin, this leads to a formula relating the Mahler measure of a non-tempered polynomial with L ′ (E , 0) , where E is an elliptic curve of conductor 20. [ABSTRACT FROM AUTHOR]