On the generators of <i>S</i>-unit groups in algebraic number fields

Brindza, Β.

doi:10.1017/s0004972700029129

Cited by 10 publications

(16 citation statements)

References 4 publications

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“…This sharpens similar inequalities due to Brindza [9] and Hajdu [14], and is of essentially the same quality as that obtained by Bugeaud and Györy [10, Lemma 1]. Of course (1.6) applies more generally to finitely generated subgroups of G that do not necessarily occur as the image of a group of S-units.…”

Section: 18])supporting

confidence: 84%

Heights on groups and small multiplicative dependencies

Vaaler

2013

Trans. Amer. Math. Soc.

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We generalize the absolute logarithmic Weil height from elements of the multiplicative group Q × /Tor Q × to finitely generated subgoups ofThe height of a finitely generated subgroup is shown to equal the volume of a certain naturally occurring, convex, symmetric subset of Euclidean space. This connection leads to a bound on the norm of integer vectors that give multiplicative dependencies among finite sets of algebraic numbers.

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Section: 18])supporting

confidence: 84%

Heights on groups and small multiplicative dependencies

Vaaler

2013

Trans. Amer. Math. Soc.

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“…The following lemma is in fact due to Hajdu [18]. It is an extended version of an earlier theorem of Brindza [5]. For convenience of the reader, we give here a proof for Lemma 1 with a slightly better value for c 4 than in [18].…”

Section: Bounds For S-units and S-regulatorsmentioning

confidence: 99%

“…To obtain these improvements we use, among other things, some recent improvements of Waldschmidt [26] and Kunrui Yu [27] concerning linear forms in logarithms, some recent estimates of Brindza [5] and Hajdu [18] for fundamental systems of S-units, some upper and lower bounds for S-regulators (cf. Lemma 3 of this paper) and an idea of Schmidt [23].…”

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confidence: 99%

Bounds for the solutions of unit equations

Bugeaud¹,

Győry

1996

Acta Arith.

121

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1. Introduction. Many diophantine problems can be reduced to (ordinary) unit equations and S-unit equations in two unknowns (for references, see e.g. [15] (with explicit constants). These led to a lot of applications.The purpose of the present paper is to considerably improve (in completely explicit form) the above-mentioned estimates in terms of the cardinality of S and of the parameters involved (degree, unit rank, regulator, class number) of the ground field. To obtain these improvements we use, among other things, some recent improvements of Waldschmidt [26] and Kunrui Yu [27] concerning linear forms in logarithms, some recent estimates of Brindza [5] and Hajdu [18] for fundamental systems of S-units, some upper and lower bounds for S-regulators (cf. Lemma 3 of this paper) and an idea of Schmidt [23]. Further, in our arguments we pay a particular attention to the dependence on the parameters in question. As a consequence of our result, we derive explicit bounds for the solutions of homogeneous linear equations of three terms in S-integers of bounded S-norm. These improve some earlier estimates of Győry [13], [14].An application of our improvements is given in [17] to decomposable form equations (including Thue equations, norm form equations and discriminant form equations) in S-integers of a number field. Some other applications will be published in two further works.

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“…It is clear that Norm L/k (u) = 1 and that u n / ∈ O * k for any n ≥ 1. By [37] (see also [9]) we may take u 0 to have logarithmic height (relative to L) h L (u 0 ) ≤ n…”

Section: 2mentioning

confidence: 99%

Counting and effective rigidity in algebra and geometry

et al. 2018

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The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.

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On the generators of S-unit groups in algebraic number fields

Abstract: Given a finitely generated multiplicative subgroup Us in a number field, we employ a simple argument from the geometry of numbers and an inequality on multiplicative dependence in number fields to obtain a minimal set of generators consisting of elements of relatively small height.

Cited by 10 publications

References 4 publications

Heights on groups and small multiplicative dependencies

Heights on groups and small multiplicative dependencies

Bounds for the solutions of unit equations

Counting and effective rigidity in algebra and geometry

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