scholarly journals Effective results for points on certain subvarieties of tori

2009 ◽  
Vol 147 (1) ◽  
pp. 69-94 ◽  
Author(s):  
ATTILA BÉRCZES ◽  
KÁLMÁN GYŐRY ◽  
JAN-HENDRIK EVERTSE ◽  
CORENTIN PONTREAU
Keyword(s):  
Number Field ◽  
Finite Rank ◽  
Upper Bounds ◽  
Accurate Description ◽  
Precise Form ◽  
Small Height ◽  
Bogomolov Conjecture ◽  
Special Classes

AbstractThe combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those pointsxof a given subvarietyof$\mathbb{G}_{\bf m}^N(\oQ )=(\oQ^*)^N$, that with respect to the height are “very close” to a given subgroup Γ of finite rank of$\mathbb{G}_{\bf m}^N(\oQ)$. Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.In this paper we prove, for certain special classes of varieties, effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the pointsxunder consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the varietywith very small height.

Number fields with large minimal index containing quadratic subfields

2018 ◽  
Vol 14 (09) ◽  
pp. 2333-2342 ◽  
Author(s):  
Henry H. Kim ◽  
Zack Wolske
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Upper Bounds ◽  
Minimal Index

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

Prime splitting in abelian number fields and linear combinations of Dirichlet characters

2014 ◽  
Vol 10 (04) ◽  
pp. 885-903 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Number Field ◽  
Nonzero Element ◽  
Number Fields ◽  
Upper Bounds ◽  
Prime Ideals ◽  
Linear Combinations ◽  
Dirichlet Characters ◽  
Abelian Number Field ◽  
Splitting Type ◽  
A Finite Group

Let 𝕏 be a finite group of primitive Dirichlet characters. Let ξ = ∑χ∈𝕏 aχ χ be a nonzero element of the group ring ℤ[𝕏]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ 𝕏 and that satisfies ∑χ∈𝕏 aχ χ(q) ≠ 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example, let K/ℚ be an abelian number field of degree n and conductor f. Let g be a proper divisor of n. If there is any unramified rational prime q that splits into g distinct prime ideals in ØK, then the least such q satisfies [Formula: see text].

scholarly journals Reducing number field defining polynomials: an application to class group computations

2016 ◽  
Vol 19 (A) ◽  
pp. 315-331
Author(s):  
Alexandre Gélin ◽  
Antoine Joux
Keyword(s):  
Number Field ◽  
Class Group ◽  
Main Idea ◽  
Class Groups ◽  
Algebraic Integers ◽  
Small Height ◽  
Subexponential Algorithm ◽  
One To One ◽  
The One

In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

scholarly journals On Effective Witt Decomposition and the Cartan–Dieudonné Theorem

2007 ◽  
Vol 59 (6) ◽  
pp. 1284-1300 ◽  
Author(s):  
Lenny Fukshansky
Keyword(s):  
Number Field ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Orthogonal Decomposition ◽  
Classical Theorem ◽  
One Dimensional ◽  
Effective Version ◽  
Special Version ◽  
Small Height ◽  
Hyperbolic Planes

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.

Estimates of lattice points in the discriminant aspect over abelian extension fields

2018 ◽  
Vol 30 (3) ◽  
pp. 767-773 ◽  
Author(s):  
Wataru Takeda ◽  
Shin-ya Koyama
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Number Fields ◽  
Upper Bounds ◽  
Abelian Extension ◽  
Lattice Points ◽  
Abelian Extensions ◽  
Lindelöf Hypothesis ◽  
Error Terms

AbstractWe estimate the number of relatively r-prime lattice points in {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

scholarly journals Bounds for the size of integral solutions to Ym = f(X)

1999 ◽  
Vol 42 (1) ◽  
pp. 127-141
Author(s):  
Dimitrios Poulakis
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Diophantine Equations ◽  
Algebraic Number Field ◽  
Upper Bounds ◽  
Ring Of Integers ◽  
Odd Order ◽  
Integral Solutions

Let K be an algebraic number field with ring of integers OK and f(X) ∈ OK[X]. In this paper we establish improved explicit upper bounds for the size of solutions in OK, of diophantine equations Y2 = f(X), where f(X) has at least three roots of odd order, and Ym = f(X), where m is an integer ≥ 3 and f(X) has at least two roots of order prime to m.

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.

NORM-EUCLIDEAN CYCLIC FIELDS OF PRIME DEGREE

2012 ◽  
Vol 08 (01) ◽  
pp. 227-254 ◽  
Author(s):  
KEVIN J. McGOWN
Keyword(s):  
Number Field ◽  
Simple Algorithm ◽  
Upper Bounds ◽  
Computational Results ◽  
Prime Degree

Let K be a cyclic number field of prime degree ℓ. Heilbronn showed that for a given ℓ there are only finitely many such fields that are norm-Euclidean. In the case of ℓ = 2 all such norm-Euclidean fields have been identified, but for ℓ ≠ 2, little else is known. We give the first upper bounds on the discriminants of such fields when ℓ > 2. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results.

Pólya S3-extensions of ℚ

2018 ◽  
Vol 149 (6) ◽  
pp. 1421-1433 ◽  
Author(s):  
Abbas Maarefparvar ◽  
Ali Rajaei
Keyword(s):  
Number Field ◽  
Upper Bounds ◽  
Ring Of Integers ◽  
Cubic Fields ◽  
Galois Closures ◽  
Unit Norm

AbstractA number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

scholarly journals A note on successive coefficients of spirallike functions

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1199-1207 ◽  
Author(s):  
Ming Li
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Extremal Functions ◽  
Lower And Upper Bounds ◽  
Bieberbach Conjecture ◽  
Precise Form ◽  
Spirallike Functions ◽  
Class Of Functions

Even there were several facts to show that ||an+1(f)|-|an(f)|| ? 1 is not true for the whole class of normalised univalent functions in the unit disk with with the form f(z) = z + ??,k=2 akzk. In 1978, Leung[7] proved ||an+1(f)|-|an(f)|| is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion |an| ? n for starlike functions. Since ||an+1(f)|-|an(f)|| ? 1 implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of ||an+1(f)|-|an(f)|| for the class of spirallike functions as this class of functions is closely related to starlike functions. In this article we prove that this functional is bounded by 1 and equality occurs only for the starlike case. We are also able to give a precise form of extremal functions. Furthermore we also try to find the sharp bound of ||an+1(f)|-|an(f)|| for non-starlike spirallike functions. By using the Carath?odory-Toeplitz theorem, we obtain the sharp lower and upper bounds of |an+1(f)|-|an(f)| for n = 1 and n = 2. These results disprove the expected inequality ||an+1(f)|-|an(f)||? cos ? for ?-spirallike functions.

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