Related papers: Mahler's method in several variables I: The theory…
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…
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…
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.
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…
This paper is devoted to the study of the analytic properties of Mahler systems at 0. We give an effective characterisation of Mahler systems that are regular singular at 0, that is, systems which are equivalent to constant ones. Similar…
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…
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…
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…
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…
Though Mahler equations have been introduced nearly one century ago, the study of their solutions is still a fruitful topic for research. In particular, the Galois theory of Mahler equations has been the subject of many recent papers.…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently,…
The paper deals with the decoupling problem of general quasilinear first order systems in two independent variables. We consider either the case of homogeneous and autonomous systems or the one of nonhomogeneous and/or nonautonomous…
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…
Mahler equations arise in a wide range of contexts including the study of finite automata, regular sequences, algebraic series over Fp(z), and periods of Drinfeld modules. Introduced a century ago by K. Mahler to study the transcendence of…
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…
We consider initial value problems of nonlinear dynamical systems, which include physical parameters. A quantity of interest depending on the solution is observed. A discretisation yields the trajectories of the quantity of interest in many…
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…
We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables.…
In mathematics and engineering, control theory is concerned with the analysis of dynamical systems through the application of suitable control inputs. One of the prominent problems in control theory is controllability which concerns the…