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Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 10 (3 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Composition factors from the group ring and Artin's theorem on orders of simple groups
 Proc. London Math. Soc
, 1990
"... The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. Th ..."
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Cited by 8 (2 self)
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The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.
Transcendence of Periods: The State of the Art
, 2005
"... function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89 Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are “interesting”, which appear “naturally”, which deserve our attention, form a countabl ..."
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Cited by 8 (0 self)
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function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89 Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are “interesting”, which appear “naturally”, which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M. Kontsevich and D. Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers. Acknowledgment This is an english updated version of the paper in french: Transcendance de périodes: état des connaissances,
3.3 Linear forms
, 2010
"... 3.3.1 Siegel’s method: m +1 linear forms For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simult ..."
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3.3.1 Siegel’s method: m +1 linear forms For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coefficients by H(L) = max{a0,..., am}. Lemma 17. Let ϑ1,..., ϑm be complex numbers. Assume that, for any ɛ> 0, there exists m +1 linearly independent linear forms L0,..., Lm in m +1 variables, with coefficients in Z, such that max 0≤k≤m Lk(1, ϑ1,..., ϑm)  < ɛ where H = max
2.3 Linear forms 2.3.1 Siegel’s method: m +1 linear forms
"... For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independe ..."
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For proving linear independence of real numbers, Hermite [6] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [14] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coefficients by H(L) = max{a0,..., am}. Lemma 13. Let ϑ1,..., ϑm be complex numbers. Assume that, for any ɛ> 0, there exists m +1 linearly independent linear forms L0,..., Lm in m +1 variables, with coefficients in Z, such that max 0≤k≤m Lk(1,ϑ1,..., ϑm)  < ɛ where H = max
1. Historical introduction to irrationality
, 2007
"... Irrationality of π by Lambert. Continued fraction expansion of e by Euler. 1.2 Fourier. Fourier’s proof of the irrationality of e. Extensions by Liouville. 1.3 Introduction to Hermite’s work. Proof of the irrationality of e r for r a nonzero rational number. Proof of the irrationality of π. Proof o ..."
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Irrationality of π by Lambert. Continued fraction expansion of e by Euler. 1.2 Fourier. Fourier’s proof of the irrationality of e. Extensions by Liouville. 1.3 Introduction to Hermite’s work. Proof of the irrationality of e r for r a nonzero rational number. Proof of the irrationality of π. Proof of the irrationality of e r ̸ ∈ Q(i) for r ∈ Q(i) ×. 2. Historical introduction to transcendence methods.
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.
The main reference is Nesterenko’s recent book [11].
, 2010
"... Proposition 1. Let ϑ be a real number. The following conditions are equivalent (i) ϑ is irrational. (ii) For any ɛ> 0, there exists p/q ∈ Q such that ..."
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Proposition 1. Let ϑ be a real number. The following conditions are equivalent (i) ϑ is irrational. (ii) For any ɛ> 0, there exists p/q ∈ Q such that