Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif

Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif

We establish new measures of linear independence of logarithms on a commutative algebraic group. Let G be a connected commutative algebraic group over Q−− and let tG be the tangent space at the origin. We consider a vector u∈tG⊗Q−C such that its image by the exponential map of the Lie group G(C) is an algebraic point p∈G(Q−−). Let V be a hyperplane in tG. We obtain lower bounds for the distance d(u,V) between u and V⊗Q−C in the rational case, where V=tH is the tangent space at the origin of an algebraic connected subgroup of G. These lower bounds are the best currently known in terms of the height h(p) of p. They generalize measures of linear forms in logarithms previously obtained by Gaudron. Our approach is based on new arguments which allow us to exclude the so-called periodic case in the demonstration, by revisiting previous work of Bertrand and Philippon. Our proofs also rely on tools from Bost’s slope theory of hermitian vector bundles. Moreover, we present ultrametric analogues of our results, and we deal with the case where V=tH is a linear subspace of any dimension.

This exploration of the relation between periods and transcendental numbers brings Baker's theory of linear forms in logarithms into its most general framework, the theory of 1-motives. Written by leading experts in the field, it contains original results and finalises the theory of linear relations of 1-periods, answering long-standing questions in transcendence theory. It provides a complete exposition of the new theory for researchers, but also serves as an introduction to transcendence for graduate students and newcomers. It begins with foundational material, including a review of the theory of commutative algebraic groups and the analytic subgroup theorem as well as the basics of singular homology and de Rham cohomology. Part II addresses periods of 1-motives, linking back to classical examples like the transcendence of π, before the authors turn to periods of algebraic varieties in Part III. Finally, Part IV aims at a dimension formula for the space of periods of a 1-motive in terms of its data.

This study investigates numbers that are powers of two and can be expressed as the sum of the squares of any two Pell numbers. We apply Baker’s theory of linear forms in logarithms of algebraic numbers, along with a variation of the Baker–Davenport reduction method, to solve the Diophantine equation [Formula: see text], where [Formula: see text], and [Formula: see text] are non-negative integers, as presented here. Furthermore, the maple[Formula: see text] codes used in the computations made throughout the paper are also shared.

On the practical solution of the Thue equation

This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

The theory of linear forms in logarithms goes back to the seventh of Hilbert’s famous problems which Hilbert stated in 1900. This problem was solved in 1934 independently by Gelfond and Schneider. They proved namely that for algebraic α, β with α ≠ 0,1 and β irrational the number γ = αβ is transcendental. This statement is equivalent to the following qualitative statement. Suppose that α, β, γ are algebraic numbers and that log α, log γ are defined and not zero. Then if Λ =β log α − log γ satisfies Λ = 0 we have β ∈ ℚ.

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

In 1939, Erdös and Mahler [‘Some arithmetical properties of the convergents of a continued fraction’, J. Lond. Math. Soc. (2)14 (1939), 12–18] studied some arithmetical properties of the convergents of a continued fraction. In particular, they raised a conjecture related to continued fractions and Liouville numbers. In this paper, we shall apply the theory of linear forms in logarithms to obtain a result in the direction of this problem.

We prove certain results pertaining to some isomorphism properties of commutative modular group algebras and briefly review a paper by pointing out some obvious mistakes and essential incorrectness.

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $d\in \mathbb {N}$ , a square matrix $A\in \mathbb {Q}^{d\times d}$ , and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets $S,T\subseteq \mathbb {R}^{d}$ , and the query is whether there exists $n\in {\mathbb {N}}$ and x ∈ S such that Anx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of $\mathbb {R}^{d}$ for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker's theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Recently G. Wustholz [5], [6] proved a theorem in transcendence which includes and greatly extends many classical results. In particular it generalizes Baker’s famous theorem [2] on linear forms in logarithms, and places it within the context of arbitrary commutative group varieties.

On the number of solutions of binomial Thue inequalities

Pillai's conjecture revisited