scholarly journals Counting Bounded Elements of a Number Field

2020 ◽  
Author(s):  
Mikołaj Fraczyk ◽  
Gergely Harcos ◽  
Péter Maga
Keyword(s):  
Group Theory ◽  
Number Field ◽  
Geometry Of Numbers ◽  
Combine Group ◽  
Successive Minima ◽  
Ideal Lattices ◽  
Linearly Independent ◽  
Ramification Theory

Abstract We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices. Our arguments combine group theory, ramification theory, and the geometry of numbers.

scholarly journals An elementary proof of Minkowski's second inequality

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Content Volume ◽  
Lattice Points ◽  
The Body ◽  
Dimensional Euclidean Space ◽  
Open Convex ◽  
Successive Minima ◽  
Linearly Independent ◽  
Independent Lattice

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

scholarly journals Three proofs of Minkowski's second inequality in the geometry of numbers

1965 ◽  
Vol 5 (4) ◽  
pp. 453-462 ◽  
Author(s):  
R. P. Bambah ◽  
Alan Woods ◽  
Hans Zassenhaus
Keyword(s):  
Convex Set ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Open Convex ◽  
Positive Real ◽  
Original Proof ◽  
Linearly Independent ◽  
Image Position

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

scholarly journals Networks and Groups in Southeast Asia: Some Observations on the Group Theory of Politics

1973 ◽  
Vol 67 (1) ◽  
pp. 103-127 ◽  
Author(s):  
Carl H. Landé
Keyword(s):  
Group Theory ◽  
Southeast Asia ◽  
Interest Group ◽  
Political Development ◽  
The Philippines ◽  
Southeast Asian ◽  
Political Structure ◽  
Combine Group ◽  
Ethnolinguistic Group ◽  
Trait Associations

The paper describes a “dyadic” type of political structure which, it is argued, is a necessary supplement to class and interest group models for the analysis of informal political structure in contemporary Southeast Asia, and probably in other developing areas.Various types of simple and complex dyadic structures are described. The paper then examines four Southeast Asian polities, of different degrees of political development, with attention to the manner in which they combine group and dyadic structures. The examples are the Kalinga, a pagan ethnolinguistic group of Northern Luzon; the Tausug, a Muslim group of the Sulu archipelago; the traditional Thai monarchy; and the present Republic of the Philippines. In each case the effects of structure upon the operation of the system are explored. The paper concludes with a set of paired propositions concerning the characteristics of “trait associations” and “personal followings.”

Variations of Minkowski's theorem on successive minima

2016 ◽  
Vol 28 (2) ◽  
Author(s):  
Martin Henk ◽  
Matthias Henze ◽  
María A. Hernández Cifre
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Geometry Of Numbers ◽  
Successive Minima

AbstractMinkowski's second theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of an

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

1991 ◽  
Vol 43 (2) ◽  
pp. 325-329 ◽  
Author(s):  
B. Brindza
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Number Fields ◽  
Finitely Generated ◽  
Geometry Of Numbers ◽  
Algebraic Number Fields ◽  
Minimal Set ◽  
Simple Argument ◽  
Small Height ◽  
Multiplicative Subgroup

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.

scholarly journals A generalisation of Minkowski's second inequality in the geometry of numbers

1966 ◽  
Vol 6 (2) ◽  
pp. 148-152 ◽  
Author(s):  
A. C. Woods
Keyword(s):  
Convex Set ◽  
Discrete Point ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Open Convex ◽  
Positive Real ◽  
Linearly Independent ◽  
Point Set

Let K be a bounded open convex set in euclidean n-space Rn symmetric in the origin 0. Further let L be a discrete point set in Rn containing 0 and at least n linearly independent points of Rn. Put mi = inf ui extended over all positive real numbers ui for which uiK contains i linearly independent points of L, i = 1, 2, …, n.

scholarly journals Linear Independence of -Logarithms over the Eisenstein Integers

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Peter Bundschuh ◽  
Keijo Väänänen
Keyword(s):  
Number Field ◽  
Linear Independence ◽  
Algebraic Number Field ◽  
Meromorphic Continuation ◽  
Quadratic Number ◽  
Arbitrary Integer ◽  
Quadratic Number Field ◽  
Linearly Independent ◽  
Imaginary Quadratic Number ◽  
Fixed Complex

For fixed complex with , the -logarithm is the meromorphic continuation of the series , into the whole complex plane. If is an algebraic number field, one may ask if are linearly independent over for satisfying . In 2004, Tachiya showed that this is true in the Subcase , , , and the present authors extended this result to arbitrary integer from an imaginary quadratic number field , and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if is the Eisenstein number field , an integer from , and a primitive third root of unity. Under these conditions, the linear independence holds also for , and both results are quantitative.

Limits of Lattices in a Compactly Generated Group

1960 ◽  
Vol 12 ◽  
pp. 427-437 ◽  
Author(s):  
A. M. Macbeath ◽  
S. Świerczkowski
Keyword(s):  
Discrete Subgroup ◽  
Dimensional Vector ◽  
Geometry Of Numbers ◽  
Linear Transformations ◽  
Bounded Lattice ◽  
Locally Compact ◽  
Dimensional Vector Space ◽  
Linearly Independent ◽  
Usual Topology ◽  
Compactly Generated

Let G be a locally compact and (σ-compact topological group and let H be a discrete subgroup of G. We shall use G/H to denote the space of right cosets Hx of H with the usual topology (cf. (8, pp. 26-28)). Let μ be the left Haar measure in G. μ induces a measure in the space G/H3; this measure will, without ambiguity in this paper, also be denoted by μ. If μ(G/H) is finite, the group H is called a lattice. If the space G/H is compact, then H is certainly a lattice and is called a bounded lattice. These terms are an extension of the usage of the Geometry of Numbers, where G is the real n-dimensional vector space Rn. In this case any lattice is generated by n linearly independent vectors, all lattices are bounded, and the whole family of lattices is permuted transitively by the automorphisms of G (which are the non-singular linear transformations).

scholarly journals Short Principal Ideal Problem in multicubic fields

2020 ◽  
Vol 14 (1) ◽  
pp. 359-392
Author(s):  
Andrea Lesavourey ◽  
Thomas Plantard ◽  
Willy Susilo
Keyword(s):  
Number Field ◽  
Algebraic Structure ◽  
Principal Ideal ◽  
Recent Progress ◽  
Number Fields ◽  
Ideal Lattices ◽  
Euclidean Lattices ◽  
Mathematical Problems ◽  
Classical Computing ◽  
Quantum Cryptosystem

AbstractOne family of candidates to build a post-quantum cryptosystem upon relies on euclidean lattices. In order to make such cryptosystems more efficient, one can consider special lattices with an additional algebraic structure such as ideal lattices. Ideal lattices can be seen as ideals in a number field. However recent progress in both quantum and classical computing showed that such cryptosystems can be cryptanalysed efficiently over some number fields. It is therefore important to study the security of such cryptosystems for other number fields in order to have a better understanding of the complexity of the underlying mathematical problems. We study in this paper the case of multicubic fields.

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