Mahler's method in several variables and finite automata
Boris Adamczewski, Colin Faverjon
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52 pages. This paper is a major rewriting of the two unpublished preprints arXiv:1809.04826 and arXiv:1809.04823 posted on the arXiv by the authors in 2018. The results obtained here far surpass the main results of these two preprints
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 Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018.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 concerning expansions of numbers in multiplicatively independent bases. For instance, we prove that no irrational real number can be automatic in two multiplicatively independent integer bases, and we give a new proof and a broad algebraic generalization of Cobham's theorem in automata theory. We also provide a new proof and a multivariate generalization of Nishioka's theorem, a landmark result in Mahler's method.
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…
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.
Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem-Mahler-Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological examples. We will state and prove a correct analog of the Skolem-Mahler-Lech theorem in positive characteristic. The zeroes of a recurrence sequence in positive characteristic can be described using finite automata.
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.
We prove several identities relating three-variable Mahler measures to integrals of inverse trigonometric functions. After deriving closed forms for most of these integrals, we obtain ten explicit formulas for three-variable Mahler measures. Several of these results generalize formulas due to Condon and Lalín. As a corollary, we also obtain three $q$-series expansions for the dilogarithm.
New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the $_5F_4$ hypergeometric function. We use these results to obtain two explicit $_5F_4$ evaluations, and several new formulas for $1/π$.
The Mahler measures of some n-variable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet L-series, generalizing the results of \cite{L}. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions.
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 $M$ variables our result depends on $\mathbb{Z}^M$ as an ordered group, and in general our lower bound depends on the choice of ordering.
The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial.
We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of 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.
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.
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.
The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to special values of L-functions of arithmetic objects (e.g. Dirichlet characters and elliptic curves over $\mathbb{Q}$). Inspired by work of Deninger Boyd has investigated this relationship numerically. In this paper we reduce some conjectures of Boyd to Beilinson`s conjectures on special values of L-functions. The methods in use are widely of K-theoretical nature…
In this paper we prove that the Mahler measures of the Laurent polynomials $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k$, $(x+x^{-1})^2(y+y^{-1})^2(1+z)^3z^{-2}-k$, and $x^4+y^4+z^4+1+k^{1/4}xyz$, for various values of $k$, are of the form $r_1 L'(f,0)+r_2 L'(χ,-1)$, where $r_1,r_2\in \mathbb{Q}$, $f$ is a CM newform of weight 3, and $χ$ is a quadratic character. Since it has been proved that these Maher measures can also be expressed in terms of logarithms and $_5F_4$-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.
We present another example of a 3-variable polynomial defining a K3-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet L-series.
In a recent work [3], the authors established new results about general linear Mahler systems in several variables from the perspective of transcendental number theory, such as a multivariate extension of Nishioka's theorem. Working with functions of several variables and with different Mahler transformations leads to a number of complications, including the need to prove a general vanishing theorem and to use tools from ergodic Ramsey theory and Diophantine approximation (e.g., a variant of the $p$-adic Schmidt subspace theorem). These complications make the proof of the main results proved i…
Given a $k$-variable Laurent polynomial $F$, any $l\times k$ integer matrix $A$ naturally defines an $l$-variable Laurent polynomial $F_A.$ I prove that for fixed $F$ the set $\mathcal M(F)$ of all the logarithmic Mahler measures $m(F_A)$ of $F_A$ for all $A$ is a closed subset of the real line. Moreover, the matrices $A$ can be assumed to be of a special form, which I call Primitive Hermite Normal Form. Furthermore, if $F$ has integer coefficients and $\mathcal M(F)$ contains $0,$ then $0$ is an isolated point of this set. I also show that, for a given bound $B>0$, the set ${\mathcal M}_B$ of…
We study the Mahler measure of the three-variable Laurent polynomial x + 1/x + y + 1/y + z + 1/z - k where k is a parameter. The zeros of this polynomial define (after desingularization) a family of K3-surfaces. In favorable cases, the K3-surface has Picard number 20, and the Mahler measure is related to its L-function. This was first studied by Marie-Jose Bertin. In this work, we prove several new formulas, extending the earlier work of Bertin.
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 6F5-hypergeometric series evaluated at 1.
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.
We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ 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 $0$ being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure $1$ is determined by its $k…
In a recent paper the team of Cogdell, Jorgenson and Smajlović develop infinite series representations for the logarithmic Mahler measure of a complex linear form, with 4 or more variables. We establish the case of 3 variables, by bounding an integral with integrand involving the random walk probability density $a\displaystyle\int_0^\infty tJ_0(at) \displaystyle\prod_{m=0}^2 J_0(r_m t)dt$, 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 beha…
Using the theory of o-minimality we show that the $p$-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if $f_1,...,f_n$ is a finite sequence of real analytic functions $f_i:(-1,1) \to (-1,1)$ for which $f_i(0) = 0$ and $|f_i'(0)| \leq 1$ (possibly zero), $a = (a_1,...,a_n)$ is an $n$-tuple of real numbers close enough to the origin and $H(x_1,...,x_n)$ is a real analytic function of $n$ variables, then the set $\{m \in {\mathbb N} : H (f_1^{\circ m} (a_1),...…
The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $Γ$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $Γ$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $Γ$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subva…
We continue to investigate the relation between the Mahler measure of certain two variable polynomials, the values of the Bloch--Wigner dilogarithm $D(z)$ and the values $ζ_F(2)$ of zeta functions of number fields. Specifically, we define a class $\A$ of polynomials $A$ with the property that $πm(A)$ is a linear combination of values $D$ at algebraic arguments. For many polynomials in this class the corresponding argument of $D$ is in the Bloch group, which leads to formulas expressing $πm(A)$ as a linear combination with unspecified rational coefficients of $V_F$ for certain number fields $F$…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a new technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Section 2 and 3, using the technique together with Mahler's method, we can reduce the al…
We give an automata-theoretic description of the algebraic closure of the rational function field F_q(t) over a finite field, generalizing a result of Christol. The description takes place within the Hahn-Mal'cev-Neumann field of "generalized power series" over F_q. Our approach includes a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, as well as some techniques for computing in the algebraic closure; these include an adaptation to positive characteristic of Newton's algorithm for finding local expansions of plane curves. …
