scholarly journals On algebraic points of fixed degree and bounded height

2021 ◽  
Vol 65 (5) ◽  
pp. 519-525
Author(s):  
D. V. Koleda
Keyword(s):  
Spatial Distribution ◽  
Asymptotic Formula ◽  
Height Function ◽  
Algebraic Numbers ◽  
Joint Work ◽  
Fixed Degree ◽  
Spatial Region ◽  
Height Functions ◽  
Distribution Of Points ◽  
Conjugate Algebraic Numbers

We consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed de- gree and bounded height. In the article the main result of a recent joint work by the author and F. Götze, and D. N. Zaporozhets is extended to the case of arbitrary height functions. We prove an asymptotic formula for the number of such algebraic points lying in a given spatial region. We obtain an explicit expression for the density function of algebraic points under an arbitrary height function.

scholarly journals On distribution densities of algebraic points under different height functions

2021 ◽  
Vol 65 (6) ◽  
pp. 647-653
Author(s):  
D. V. Koleda
Keyword(s):  
Distribution Density ◽  
Height Function ◽  
Upper And Lower Bounds ◽  
Random Polynomial ◽  
Algebraic Numbers ◽  
Distribution Of Zeros ◽  
Fixed Degree ◽  
Height Functions ◽  
Distribution Of Points ◽  
Conjugate Algebraic Numbers

In the article we consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed degree. The distribution is introduced using a height function. We have obtained universal upper and lower bounds of the distribution density of such points using an arbitrary height function. We have shown how from a given joint density function of coefficients of a random polynomial of degree n, one can construct such a height function H that the polynomials q of degree n uniformly chosen under H[q] ≤1 have the same distribution of zeros as the former random polynomial.

scholarly journals The distribution of close conjugate algebraic numbers

2010 ◽  
Vol 146 (5) ◽  
pp. 1165-1179 ◽  
Author(s):  
Victor Beresnevich ◽  
Vasili Bernik ◽  
Friedrich Götze
Keyword(s):  
Diophantine Approximation ◽  
Real Line ◽  
Algebraic Numbers ◽  
Fixed Degree ◽  
Irreducible Polynomials ◽  
The Real ◽  
Conjugate Algebraic Numbers

AbstractWe investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

scholarly journals On the frequency of height values

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Gabriel A. Dill
Keyword(s):  
Unit Disk ◽  
Counting Function ◽  
Height Function ◽  
Dynamical Behaviour ◽  
Fixed Number ◽  
Open Unit ◽  
Open Unit Disk ◽  
Order Of Growth ◽  
Algebraic Numbers ◽  
Fixed Degree

AbstractWe count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height $${\mathcal {H}}$$ H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to $${\mathcal {H}}$$ H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if $$k \in \{0,d\}$$ k ∈ { 0 , d } or $$\gcd (k,d) = 1$$ gcd ( k , d ) = 1 . We therefore study the behaviour in the case where $$0< k < d$$ 0 < k < d and $$\gcd (k,d) > 1$$ gcd ( k , d ) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.

scholarly journals Joint distribution of conjugate algebraic numbers: A random polynomial approach

2020 ◽  
Vol 359 ◽  
pp. 106849 ◽  
Author(s):  
Friedrich Götze ◽  
Denis Koleda ◽  
Dmitry Zaporozhets
Keyword(s):  
Joint Distribution ◽  
Random Polynomial ◽  
Algebraic Numbers ◽  
Conjugate Algebraic Numbers

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

2010 ◽  
Vol 06 (03) ◽  
pp. 471-499 ◽  
Author(s):  
EVELINA VIADA
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Abelian Variety ◽  
Lower Bounds ◽  
Finite Rank ◽  
Height Function ◽  
Algebraic Numbers ◽  
Algebraic Subgroup ◽  
Rank Zero ◽  
The Third

This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non-Zariski-dense in V. If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.

On spacelike curves in hyperbolic space times sphere

2014 ◽  
Vol 11 (03) ◽  
pp. 1450014 ◽  
Author(s):  
Lingling Kong ◽  
Donghe Pei
Keyword(s):  
Hyperbolic Space ◽  
Height Function ◽  
Point Of View ◽  
De Sitter ◽  
Height Functions ◽  
Geometric Point

The main goal of this paper is to study singularities of lightlike surfaces and focal surfaces of spacelike curves in Hyperbolic space times sphere. To do that, we construct a de Sitter height function and a Lightcone height function, and then show the relation between singularities of the lightlike surfaces (respectively, the focal surfaces) and that of the de Sitter height functions (respectively, the Lightcone height functions). In addition, some geometry properties of the spacelike curves are studied from geometric point of view.

Measurement of feature clustering in images

1990 ◽  
Vol 48 (1) ◽  
pp. 552-553
Author(s):  
John C. Russ
Keyword(s):  
Spatial Distribution ◽  
Random Distribution ◽  
Nearest Neighbor ◽  
Mean Value ◽  
Feature Clustering ◽  
Spatial Coordinates ◽  
Distribution Of Points ◽  
The Mean ◽  
The Individual ◽  
Nearest Neighbor Distances

The spatial distribution of features in an image is often interesting, but not simple to characterize. Mapping of the image into a different space (e.g., Fourier or Hough) offers direct information on various regularities in feature spacing or alignment, but does not deal directly with the individual features. Two other approaches are available; each has advantages and drawbacks, which are discussed here.Schwarz & Exner determine the spatial coordinates of the centroids of features, and sort them to locate the nearest neighbor for each feature present, constructing a distribution plot of the frequency of nearest neighbor distances. Figures 1 and 2 show an example. The three fields in Figure 1 contain, respectively, features which are well-spaced from each other, randomly arranged on the plane, and clustered together. For the random distribution of points, the histogram of nearest neighbor distances is a Poisson distribution, and the mean value is 0.5/NA1/2, where NA is the number of features divided by the area of the image.

Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields

2007 ◽  
Vol 59 (1) ◽  
pp. 186-210 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto
Keyword(s):  
Quadratic Function ◽  
Height Function ◽  
Rational Functions ◽  
Closed Field ◽  
Height Functions ◽  
Planar Curve ◽  
Endomorphism Algebras ◽  
Form Part ◽  
Quadratic Function Field ◽  
Quadratic Function Fields

AbstractPurely simple Kronecker modules ℳ, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h: K ∪ ﹛∞﹜ → ﹛∞, 0, 1, 2, 3, … ﹜ is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X)[Y] called the regulator. When the module ℳ admits nontrivial endomorphisms, f must be linear or quadratic in Y. In that case ℳ is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End ℳ embeds in the quadratic function field K(X)[Y]/(f). For some height functions h of infinite support I, the search for a functional α for which (h, α) has regulator 0 comes down to having functions η : I → K such that no planar curve intersects the graph of η on a cofinite subset. If K has characterictic not 2, and the triplet (m, h, α) gives a purely-simple Kronecker module ℳ having non-trivial endomorphisms, then h attains the value ∞ at least once on K ∪ ﹛∞﹜ and h is finite-valued at least twice on K ∪ ﹛∞﹜. Conversely all these h form part of such triplets. The proof of this result hinges on the fact that a rational function r is a perfect square in K(X) if and only if r is a perfect square in the completions of K(X) with respect to all of its valuations.

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