An upper bound on the number of perfect quadratic forms

Woerden, Wessel P. J. van

doi:10.1016/j.aim.2020.107031

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…Classifying perfect matrices up to arithmetical equivalence had already attracted Lagrange (n = 2), Gauß (n = 3) and others. Based on his theory Voronoi gave an algorithm for the classification for any given dimension n. Still being the only known approach for classification in arbitrary dimension, it has been used by several authors, see [8], [18], [24], [27]. Due to the difficulty of polyhedral representation conversions the classification is nevertheless still open for all n ≥ 9.…”

Section: Perfect Positive Definite Matricesmentioning

confidence: 99%

Perfect Copositive Matrices

Valentin¹,

Schürmann²

2023

Communications in Mathematics

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In this paper we give a first study of perfect copositive $n \times n$ matrices. They can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive definite matrices. Most of the differences occur only for $n \geq 3$, where we find for instance lower rank and indefinite perfect matrices. Nevertheless, we find for all $n$ that for every classical perfect matrix there is an arithmetically equivalent one which is also perfect copositive. Furthermore we study the neighborhood graph and polyhedral structure of perfect copositive matrices. As an application we obtain a new characterization of the cone of completely positive matrices: It is equal to the set of nonnegative matrices having a nonnegative inner product with all perfect copositive matrices.

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Section: Perfect Positive Definite Matricesmentioning

confidence: 99%

Perfect Copositive Matrices

Valentin¹,

Schürmann²

2023

Communications in Mathematics

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“…In this paper, we follow a similar argument to van Woerden in [6] in order to determine an upper bound on the number of homothety classes of perfect unary forms for an arbitrary totally real number field. Our result is stated as follows.…”

Section: Introductionmentioning

confidence: 99%

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

Porter¹,

Mendelsohn²

2022

Preprint

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Let K be a totally real number field of degree n over Q, with discriminant and regulator ∆ K , R K respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above byMoreover, if K is a unit reducible field, the number of classes of perfect unary forms is bound above by O(∆ K exp(2n log(n))).

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An upper bound on the number of classes of perfect unary forms in totally real number fields

Porter,

Mendelsohn

2024

Int. J. Number Theory

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Let [Formula: see text] be a totally real number field of degree [Formula: see text] over [Formula: see text], with discriminant and regulator [Formula: see text], respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by [Formula: see text], where [Formula: see text] is the discriminant of the field [Formula: see text], [Formula: see text] is the additive Hermite–Humbert constant over positive-definite unary forms for [Formula: see text] and [Formula: see text] is the covering radius of the log-unit lattice. In particular, when [Formula: see text] is Galois over [Formula: see text] and [Formula: see text] is a prime number, the number of homothety classes of unary forms is upper bounded by [Formula: see text], where [Formula: see text] is the regulator of [Formula: see text]. Moreover, if [Formula: see text] is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by [Formula: see text].

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An upper bound on the number of perfect quadratic forms

Abstract: In a recent preprint on arXiv Roland Bacher showed that the number p d of non-similar perfect d-dimensional quadratic forms satisfies e Ω(d) < p d < e O(d 3 log(d)) . We improve the upper bound to e O(d 2 log(d)) by a volumetric argument based on Voronoi's first reduction theory.

Cited by 3 publications

References 13 publications

Perfect Copositive Matrices

Perfect Copositive Matrices

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

An upper bound on the number of classes of perfect unary forms in totally real number fields

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