Boris Adamczewski
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View article: Algebraic Independence Measures for Values of E-functions and M-functions
Algebraic Independence Measures for Values of E-functions and M-functions Open
In this article, we establish a Liouville-type inequality for polynomials evaluated at the values of arbitrary Siegel E-functions at non-zero algebraic points. Additionally, we provide a comparable result within the framework of Mahler M -…
View article: Addendum to: Mahler's method in several variables and finite automata
Addendum to: Mahler's method in several variables and finite automata Open
This note is an addendum to the paper ''Mahler's method in several variables and finite automata''. It strengthens part (i) of Theorem 1.1 of the aforementioned paper.
View article: Algebraic relations between sine and cosine values
Algebraic relations between sine and cosine values Open
The aim of this note is to show that any algebraic relation over $\overline{\mathbb{Q}}$ between the values of the trigonometric functions sine and cosine at algebraic points can be derived from the Pythagorean identity and the angle addit…
View article: Cyclotomic valuation of $q$-Pochhammer symbols and $q$-integrality of basic hypergeometric series
Cyclotomic valuation of $q$-Pochhammer symbols and $q$-integrality of basic hypergeometric series Open
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View article: On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words
On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words Open
We consider numbers of the form S_β(u): = ∑_{n=0}^∞ (u_n)/(βⁿ), where u = ⟨u_n⟩_{n=0}^∞ is an infinite word over a finite alphabet and β ∈ ℂ satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echo…
View article: A new proof of Nishioka’s theorem in Mahler’s method
A new proof of Nishioka’s theorem in Mahler’s method Open
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 f…
View article: A sharper multivariate Christol's theorem with applications to diagonals and Hadamard products
A sharper multivariate Christol's theorem with applications to diagonals and Hadamard products Open
We provide a new proof of the multivariate version of Christol's theorem about algebraic power series with coefficients in finite fields, as well as of its extension to perfect ground fields of positive characteristic obtained independentl…
View article: Bracket words: A generalisation of Sturmian words arising from generalised polynomials
Bracket words: A generalisation of Sturmian words arising from generalised polynomials Open
Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials. …
View article: Relations algébriques entre valeurs de E-fonctions ou de M-fonctions
Relations algébriques entre valeurs de E-fonctions ou de M-fonctions Open
We prove that all algebraic relations over $\overline{\mathbb Q}$ between values of Siegel's $E$-functions at some non-zero algebraic point have a functional source, in that they can be obtained as degeneration of $δ$-algebraic relations o…
View article: Algebraic independence and linear difference equations
Algebraic independence and linear difference equations Open
We consider pairs of automorphisms (\phi,\sigma) acting on fields of Laurent or Puiseux series: pairs of shift operators (\phi\colon x\mapsto x+h_1, \sigma\colon x\mapsto x+h_2) , of q -difference operators (\phi\colon x\mapsto q_1x , \sig…
View article: A new proof of Nishioka's theorem in Mahler's method
A new proof of Nishioka's theorem in Mahler's method Open
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 f…
View article: A height gap theorem for coefficients of Mahler functions
A height gap theorem for coefficients of Mahler functions Open
We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, ther…
View article: (Logarithmic) densities for automatic sequences along primes and squares
(Logarithmic) densities for automatic sequences along primes and squares Open
In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic seque…
View article: Mahler's method in several variables and finite automata
Mahler's method in several variables and finite automata Open
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 independen…
View article: Hypertranscendence and linear difference equations
Hypertranscendence and linear difference equations Open
After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (i.e., they cannot be solution to an algebraic…
View article: A height gap theorem for coefficients of Mahler functions
A height gap theorem for coefficients of Mahler functions Open
We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, ther…
View article: On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited
On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited Open
We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis--Stearns problem can be solved in a satisfactory way for the class of multistack machines.…
View article: A note on Christol's theorem
A note on Christol's theorem Open
Christol's theorem characterises algebraic power series over finite fields in terms of finite automata. In a recent article, Bridy develops a new proof of Christol's theorem by Speyer, to obtain a tight quantitative version, that is, to bo…
View article: Mahler's method in several variables II: Applications to base change problems and finite automata
Mahler's method in several variables II: Applications to base change problems and finite automata Open
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 g…
View article: Mahler's method in several variables I: The theory of regular singular systems
Mahler's method in several variables I: The theory of regular singular systems Open
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…
View article: Exceptional values of E-functions at algebraic points
Exceptional values of E-functions at algebraic points Open
E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siege…
View article: Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series
Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series Open
We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-anal…
View article: M\\'ethode de Mahler, transcendance et relations lin\\'eaires : aspects\n effectifs
M\\'ethode de Mahler, transcendance et relations lin\\'eaires : aspects\n effectifs Open
This note deals with some effective results in Mahler's method. In a recent\nwork, we used a theorem of Philippon to show that given a Mahler function\n$f(z)$ in ${\\bf k}\\{z\\}$, where ${\\bf k}$ denotes a number field, and an\nalgebraic…
View article: Algebraic independence of $G$-functions and congruences "à la Lucas"
Algebraic independence of $G$-functions and congruences "à la Lucas" Open
We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite …
View article: On the computational complexity of algebraic numbers: the\n Hartmanis--Stearns problem revisited
On the computational complexity of algebraic numbers: the\n Hartmanis--Stearns problem revisited Open
We consider the complexity of integer base expansions of algebraic irrational\nnumbers from a computational point of view. We show that the Hartmanis--Stearns\nproblem can be solved in a satisfactory way for the class of multistack\nmachin…
View article: Méthode de Mahler: relations linéaires, transcendance et applications aux nombres automatiques
Méthode de Mahler: relations linéaires, transcendance et applications aux nombres automatiques Open
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…