Chris Smyth
A survey of results for Mahler measure of algebraic numbers, and one-variable 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 given too, though not to Mahler measure of polynomials in more than one variable.
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Here we propose a survey on Mahler's theory for transcendence and algebraic independence focusing on certain applications to the arithmetic of periods of Anderson t-motives.
In this note we prove algebraic independence results for the values of a special class of Mahler functions. In particular, the generating functions of Thue-Morse, regular paperfolding and Cantor sequences belong to this class, and we obtain the algebraic independence of the values of these functions at every non-zero algebraic point in the open unit disk. The proof uses results on Mahler's method.
The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the other things, we provide a measure of algebraic independence for values of Mahler's functions at complex transcendental points, a result of type which has never appeared in the literature in the past. As an example of application of our new measures of algebraic independence, we prove that a Mahler number does not belong to the class $U$ in Mahler's classifica…
We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for $n\geq 1$ arbitrarily large, new examples of sets $(θ_1,...,θ_n)\in\mrr^n$ normal in the sense of definition formulated by Grigory Chudnovsky (1980).
Dubickas 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 as well as establish its analog for the ultrametric Mahler measure.
Transcendence of multivariate Mahler functions and algebraic relations between their values
研究多元马勒函数的函数性质，证明其在特定区域的解析性与亚纯性及有理 - 超越二分法，加强相关提升定理，能计算函数值间代数关系，还证明了正则点情况下的多元下降定理。
Here we characterise in a complete and explicit way the relations of algebraic dependence over $\QQ$ of complex values of Hecke-Mahler series taken at algebraic points of the multiplicative group $\GG_m^2(\CC)$. Our result contains previous theorems by Loxton and van der Poorten, Mahler, and Masser.
Let $\mathbb{K}$ be a function field of characteristic $p>0$. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over $\mathbb{K}(z)$. This paper is dedicated to proving the following refinement of this theorem. Let $f_{1}(z),\ldots f_{n}(z)$ be $d$-Mahler functions such that $\overline{\mathbb{K}}(z)\left(f_{1}(z),\ldots, f_{n}(z)\right)$ is a regular extension over $\overline{\mathbb{K}}(z)$. Then, every homogeneous algebraic relation over $\overline{\mathbb{K}}$ between their values at a regular algebraic point arises as the specialization of…
We study the dynamical Mahler measure of multivariate polynomials and present dynamical analogues of various results from the classical Mahler measure as well as examples of formulas allowing the computation of the dynamical Mahler measure in certain cases. We discuss multivariate analogues of dynamical Kronecker's Lemma and present some improvements on the result for two variables due to Carter, Lalín, Manes, Miller, and Mocz.
This paper is concerned with Mahler's method. We study in detail the structure of linear relations between values of Mahler functions at algebraic points. In particular, given a field ${\bf k}$, a Mahler function $f(z)\in{\bf k}\{z\}$, and an algebraic number $α$, $0<\vert α\vert <1$, that is not a pole for $f$, we show that one can always determined whether the number $f(α)$ is transcendental or not. In the latter case, we obtain that $f(α)$ belong to the number fields ${\bf k}(α)$. We also consider some consequences of such results to a classical number theoretical problem: the study of sequ…
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), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore defines a dynamical system. The \emph{orbit size} of $α$, denoted $\# \mathcal{O}_M(α)$, 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 exis…
Mahler's method and Carlitz logarithm
本文用马勒方法重新证明了卡尔利茨对数在有理函数域上线性无关则代数无关的定理，扩展了Denis的方法。
Some results on asymptotic versions of Mahler's problems
该研究属于数论领域，证明了两类关于系数几乎全有界的超越函数的结果，推进了Mahler问题渐近版本的相关研究。
In a previous paper, the authors introduced several vector space norms on the space of algebraic numbers modulo torsion which corresponded to the Mahler measure on a certain class of numbers and allowed the authors to formulate L^p Lehmer conjectures which were equivalent to their classical counterparts. In this paper, we introduce and study several analogous 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 evaluate these norms on ce…
In 1902, P. Stäckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.
We 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…
We show that for almost every polynomial P(x,y) with complex coefficients, the difference of the logarithmic Mahler measures of P(x,y) and P(x,x^n) can be expanded in a type of formal series similar to an asymptotic power series expansion in powers of 1/n. This generalizes a result of Boyd. We also show that such an expansion is unique and provide a formula for its coefficients. When P has algebraic coefficients, the coefficients in the expansion are linear combinations of polylogarithms of algebraic numbers, with algebraic coefficients.
Mahler's measure defines a dynamical system on the algebraic numbers. In this paper, we study the problem of which number fields have points which wander under the iteration of Mahler's measure. We completely solve the problem for all abelian number fields, and more generally, for all extensions of the rationals of degree at most five.
Let $M(α)$ denote the Mahler measure of the algebraic number $α$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar techniques to study a non-Archimedean version. We show how to generalize the above constructions in order to associate, to each point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each having a triangle inequality of a different strength. We are able to compute $M_x(α)$ for sufficiently small $x$, identifying, in the process, a function…
This note deals with some effective results in Mahler's method. In a recent work, we used a theorem of Philippon to show that given a Mahler function $f(z)$ in ${\bf k}\{z\}$, where ${\bf k}$ denotes a number field, and an algebraic number $α$ in the domain of holomorphy of $f$, the number $f(α)$ is either transcendental or belongs to ${\bf k}(α)$. We describe here an effective procedure to decide if such a number is transcendental or not. More generally, given several Mahler functions $f_1(z),\cdots,f_r(z)$ and an algebraic number $α$ in the domain of holomorphy of these functions, we show ho…
In 1902, P. Stäckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. Recently, the authors answered positively this question. In this paper, we prove a much stronger version of this result by considering other subsets of $\mathbb{C}$.
We present a new method for algebraic independence results in the context of Mahler's method. In particular, our method uses the asymptotic behaviour of a Mahler function $f(z)$ as $z$ goes radially to a root of unity to deduce algebraic independence results about the values of $f(z)$ at algebraic numbers. We apply our method to the canonical example of a degree two Mahler function; that is, we apply it to $F(z)$, the power series solution to the functional equation $F(z)-(1+z+z^2)F(z^4)+z^4F(z^{16})=0$. Specifically, we prove that the functions $F(z)$, $F(z^4)$, $F'(z)$, and $F'(z^4)$ are alg…
This is the second part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. From the lifting theorem obtained in the first part, we first derive a general result, showing that Mahler functions in several variables, associated with transformations having multiplicatively dependent spectral radii, take algebraic independent values at algebraic points provided that these points are sufficiently independent. Then, we focus on applications of this result and of the two main results of Part I of this work. Our…
A Liouville-Type Inequality for Values of Mahler M-Functions
本文基于马赫尔Mq函数在公共非零代数点处的值，建立利维勒型不等式，并证明此类值既不是利维勒数也不是U数，解决了该领域长期存在的问题。
There are many examples of several-variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger, and later Rodriguez-Villegas, and Maillot. While Rodriguez-Villegas made this relationship explicit for the two variable case, it is our goal to understand the three variable case and shed some light on the examples with more variables.
In this paper we construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is generated by a linear recurrence. In order to prove this result, we reduce the algebraic independency to that of Mahler functions of several variables by shifting the linear recurrence and apply the theory of Mahler functions.
In this paper, we study a relation between digits of $p$-adic numbers and Mahler's classification. We show that an irrational $p$-adic number whose digits are automatic, primitive morphic, or Sturmian is an $S$-, $T$-, or $U_1$-number in the sense of Mahler's classification. Furthermore, we give an algebraic independence criterion for $p$-adic numbers whose digits are Sturmian.
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.
We provide a general result for the algebraic independence of Mahler functions by a new method based on asymptotic analysis. As a consequence of our method, these results hold not only over $\mathbb{C}(z)$, but also over $\mathbb{C}(z)(\mathcal{M})$, where $\mathcal{M}$ is the set of all meromorphic functions. Several examples and corollaries are given, with special attention to nonnegative regular functions.
The asymptotic Mahler measure of Gaussian periods
构造具有特别小Mahler测度的分圆整数（高斯周期）序列，研究其Mahler测度作为导子函数的渐近行为，发现增长率是维数递增的log Calabi-Yau簇族的多元Mahler测度。
For every complex number $x$, let $\Vert x\Vert_{\mathbb{Z}}:=\min\{|x-m|:\ m\in\mathbb{Z}\}$. Let $K$ be a number field, let $k\in\mathbb{N}$, and let $α_1,\ldots,α_k$ be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of $θ\in (0,1)$ such that there are infinitely many tuples $(n,q_1,\ldots,q_k)$ satisfying $\Vert q_1α_1^n+\ldots+q_kα_k^n\Vert_{\mathbb{Z}}<θ^n$ where $n\in\mathbb{N}$ and $q_1,\ldots,q_k\in K^*$ having small logarithmic height compared to $n$. In the special case when $q_1,\ldots,q_k$ have the form $q_i=qc_i$ for fixed $c_1,\ldots,c…
This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real number $θ$ we associate its polynomial diophantine approximation ring: a tri-filtered subring of a nonstandard model of the ring $\mathbb{Z}[X]$. We characterize the filtration structure of the polynomial diophantine approximation ring according to the Mahler class and the Mahler type of $θ$. The arithmetic of polynomial diophantine approximation groups is intro…
Suppose that $F(x)\in\mathbb{Z}[[x]]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$. In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If $F(x)$ is also regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$-number or a $T$-number.
After Furstenberg had provided a first glimpse of remarkable rigidity phenomena associated with the joint action of several commuting automorphisms (or endomorphisms) of a compact abelian group, further key examples motivated the development of an extensive theory of such actions. Two of Mahler's achievements, the recognition of the significance of Mahler measure of multivariate polynomials in relating the lengths and heights of products of polynomials in terms of the corresponding quantities for the constituent factors, and his work on additive relations in fields, have unexpectedly played im…
For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ an…
Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
研究具有拉格朗日性质的动力系统和簇代数中马勒度量的增长率，比较其与其他增长度量，并提出马勒度量的增长率与代数熵一致的猜想。
The finiteness of computing the ultrametric Mahler measure
针对代数数的超度量Mahler测度$M_\infty(α)$的计算问题，作者证明其可归约为对有限集的搜索，并给出可计算的实例以确定该测度。
About fifty years ago Mahler proved that if $α>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $α^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $α$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $α$ such that the continued fraction of $α^n$ has period length tending to infinity.
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 $m(P_{a,c})$.
On the non-Archimedean metric Mahler measure
本文提出了度量Mahler测度的非阿基米德版本$M_\infty$，证明了其取值为1当且仅当$α$为单位根，并验证了其在相应向量空间上定义了射影高度，最后给出了根式情形下的计算方法。
For an algebraic number $α$, the metric Mahler measure $m_1(α)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(α)$ by the author in 2010. The definition of $m_t(α)$ involves taking an infimum over a certain collection $N$-tuples of points in $\overline{\mathbb Q}$, and from previous work of Jankauskas and the author, the infimum in the definition of $m_t(α)$ is attained by rational points when $α\in \mathbb Q$. As a consequence of our main theorem in this article, we obtain an analog of this result when $\mathbb Q$ is replaced wi…
In this work we apply the techniques that were developed in [Lalin: An algebraic integration for Mahler measure] 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 [Gon, Oyanagi: Generalized Mahler measures and multiple sine functions]
It is a classical result of Mahler that for any rational number $α$ > 1 which is not an integer and any real 0 < c < 1, the set of positive integers n such that $α$ n < c n is necessarily finite. Here for any real x, x denotes the distance from its nearest integer. The problem of classifying all real algebraic numbers greater than one exhibiting the above phenomenon was suggested by Mahler. This was solved by a beautiful work of Corvaja and Zannier. On the other hand, for non-zero real numbers $λ$ and $α$ with $α$ > 1, Hardy about a century ago asked ''In what circumstances can it be true that…
Let $θ$ and $ρ$ be real numbers with $0 \le θ, ρ< 1$ and $θ$ irrational. We show that the Hecke-Mahler series $$ F_{θ, ρ} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 θ+ ρ\rfloor} \, z_1^{k_1} z_2^{k_2}, $$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(β, α)$ with $0 < |β|, |βα^θ| < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $ρ=0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give t…
On a theorem of Garza regarding algebraic numbers with real conjugates
本文为Garza关于实共轭代数数高度估计的定理提供了一种新颖而简洁的证明。
The parametrized family of metric Mahler measures
本文推广了Mahler测度的度量版本$M_t$，利用其给出Lehmer猜想的等价形式，证明该函数由指数函数分段构造，并提出一个关于有理数图像绘制的猜想。
Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging the log of the v-adic absolute value of F over the periodic points of g. This allows us to compute canonical heights of algebraic points via equidistribution on periodic points.
The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic independence of function values to the one of functions if these satisfy certain types of functional equations.
We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent matrix transformations. Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018. The main new feature is that they apply now without any restriction on the matrices defining the corresponding Mahler systems. As a consequence, we settle several problems concern…
Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $δ$ and $ϕ$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.
Algebraic Independence Measures for Values of E-functions and M-functions
本文针对任意Siegel E函数在非零代数点处的值构成的多项式建立Liouville型不等式，并在Mahler M函数框架内给出可比较结果，推进了特殊函数值的代数无关性研究。
This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at the origin. Our interest in Mahler's method comes from the possible applications of these results to old problems involving automata theory and which concern the expansion of both natural numbers and real numbers in integer bases. In particular, problems which involve finite automata and base change. Such applications are studied in Part II of this work.
Transcendence of Hecke-Mahler Series
本文证明了Hecke-Mahler级数在特定条件下的超越性，其中涉及非常数整数多项式、无理数参数和绝对值大于1的代数数基底。
In the theory of continued fractions, Zaremba's conjecture states that there is a positive integer $M$ such that each integer is the denominator of a convergent of an ordinary continued fraction with partial quotients bounded by $M$. In this paper, to each such $M$ we associate a regular sequence---in the sense of Allouche and Shallit---and establish various properties and results concerning the generating function of the regular sequence. In particular, we determine the minimal algebraic relation concerning the generating function and its Mahler iterates.
On Mahler $U$-numbers under rational maps
本文证明具有实代数系数的有理函数将类型为$m$的Mahler $U$-数映射为类型在$\lceil m/(ed)\rceil$到$em$之间的$U$-数，且所有类型均可实现，并由此刻画了保持该类不变的仅为$\mathbb{Q}$上的Möbius变换。
We show that the Mahler measure of every Borwein polynomial -- a polynomial with coefficients in $ \{-1,0,1 \}$ having non-zero constant term -- can be expressed as a maximal Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer's problem for height-one polynomials having minimal Mahler measure becomes equivalent to a natural question from the spectral theory of binary constant-length substitutions. This supports another connection between Mahler measures and dynamics, beyond the well-known appearance of Mahler mea…
In the frame of Mahler's method for algebraic independence we show that the algebraic relations over Q linking the values of functions solutions of a system of functional equations come from the algebraic relations between the functions themselves, by specialisation. We deduce some results on the linear independence of the q-regular functions.
Schur studied limits of the arithmetic means $A_n$ of zeros for polynomials of degree $n$ with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that $\limsup_{n\to\infty} |A_n| \le 1-\sqrt{e}/2.$ We show that $A_n \to 0$, and estimate the rate of convergence by generalizing the Erdős-Turán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conj…
The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal polynomial sequence for exponential sums over finite fields. The degree periodicity also holds for some more complicated non-linear recurrence sequences. We give one example from the iterations of a polynomial map. This depending on the dynamic Mordell-Lang conjecture which has been proved in some cases.
On a version of the Problem B of Mahler involving derivatives
本文研究 Mahler 问题 B 的导数推广版本，证明存在不可数多个超越整函数，使其本身、反函数及导数在代数点上均取代数值。
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