scholarly journals A FINITENESS PROPERTY FOR PREPERIODIC POINTS OF CHEBYSHEV POLYNOMIALS

2010 ◽  
Vol 06 (05) ◽  
pp. 1011-1025 ◽  
Author(s):  
SU-ION IH ◽  
THOMAS J. TUCKER
Keyword(s):  
Number Field ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Algebraic Closure ◽  
Finiteness Property ◽  
Finite Set

Let K be a number field with algebraic closure [Formula: see text], let S be a finite set of places of K containing the Archimedean places, and let φ be a Chebyshev polynomial. We prove that if [Formula: see text] is not preperiodic, then there are only finitely many preperiodic points [Formula: see text] which are S-integral with respect to α.

scholarly journals Integral division points on curves

2013 ◽  
Vol 149 (12) ◽  
pp. 2011-2035 ◽  
Author(s):  
David Grant ◽  
Su-Ion Ih
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Abelian Varieties ◽  
Algebraic Closure ◽  
Effective Divisor ◽  
Dimensional Case ◽  
Finitely Generated ◽  
General Conjecture ◽  
Finite Set ◽  
Set Of Points

AbstractLet $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $E( \overline{k} )$’, i.e., the set of points of $\mit{\Gamma} $ which are $S$-integral on $E$ relative to $D, $ is finite. We also prove the ${ \mathbb{G} }_{\mathrm{m} } $-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

scholarly journals New Properties of Modified Second Kind Chebyshev Polynomials

2020 ◽  
Vol 55 (3) ◽  
Author(s):  
Semaa Hassan Aziz ◽  
Mohammed Rasheed ◽  
Suha Shihab
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Digital Computer ◽  
Higher Order ◽  
Algebraic Equations ◽  
Equation Problem ◽  
Higher Order Differential Equations

Modified second kind Chebyshev polynomials for solving higher order differential equations are presented in this paper. This technique, along with some new properties of such polynomials, will reduce the original differential equation problem to the solution of algebraic equations with a straightforward and computational digital computer. Some illustrative examples are included. The modified second kind Chebyshev polynomial is calculated using only a small number of the modified second kind Chebyshev polynomials, which leads to attractive results.

Chebyshev polynomials on a finite set of points

1970 ◽  
Vol 6 (12) ◽  
pp. 1372-1374
Author(s):  
V. A. Bovin
Keyword(s):  
Chebyshev Polynomials ◽  
Finite Set ◽  
Set Of Points

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

AbstractLet V be a K3 surface defined over a number field k. The Batyrev-Manin conjecture for V states that for every nonempty open subset U of V, there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU for successively smaller sets U.In this paper, in the case that V is a Kummer surface, we prove that the Batyrev-Manin conjecture for V can be reduced to the Batyrev-Manin conjecture for V modulo the endomorphisms of V induced by multiplication by m on the associated abelian surface A. As an application, we use this to show that given some restrictions on A, the set of rational points of V which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

On tamely ramified pro-p-extensions over -extensions of

2013 ◽  
Vol 156 (2) ◽  
pp. 281-294
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Rational Number ◽  
Prime Numbers ◽  
Finite Set

AbstractFor an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

The hyperelliptic equation over function fields

1983 ◽  
Vol 93 (2) ◽  
pp. 219-230 ◽  
Author(s):  
R. C. Mason
Keyword(s):  
Elliptic Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Integer Coefficients ◽  
Finite Set ◽  
Integer Solutions ◽  
Explicit Bounds

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.

Descente galoisienne sur le groupe de Brauer

2012 ◽  
Vol 2013 (682) ◽  
pp. 141-165
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Alexei N. Skorobogatov
Keyword(s):  
Number Field ◽  
Characteristic Zero ◽  
Projective Variety ◽  
Algebraic Closure ◽  
Invariant Subgroup ◽  
Brauer Group ◽  
Intersection Pairing ◽  
Nous Montrons ◽  
Numerical Equivalence

Abstract. Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur un corps de nombres, nous donnons des estimations sur l'ordre de ce quotient. L'accouplement d'intersection entre les groupes de diviseurs et de 1-cycles modulo équivalence numérique joue ici un rôle important. For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions, e.g., over a number field, we give estimates for the order of this cokernel. We emphasise the rôle played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

Fox formulas for twisted Alexander invariants associated to representations of knot groups over rings of S-integers

2018 ◽  
Vol 27 (05) ◽  
pp. 1850033 ◽  
Author(s):  
Ryoto Tange
Keyword(s):  
Number Field ◽  
Asymptotic Growth ◽  
Homology Groups ◽  
Finite Set ◽  
Twisted Homology ◽  
Alexander Invariants

We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of [Formula: see text]-integers of [Formula: see text], where [Formula: see text] is a finite set of finite primes of a number field [Formula: see text]. As an application, we give the asymptotic growth of twisted homology groups.

ON THE PROBLEM OF INTEGER SOLUTIONS TO DECOMPOSABLE FORM INEQUALITIES

2008 ◽  
Vol 04 (05) ◽  
pp. 859-872 ◽  
Author(s):  
YUANCHENG LIU
Keyword(s):  
Real Number ◽  
Number Field ◽  
Finite Set ◽  
Integer Solutions ◽  
Decomposable Form

This paper proves a conjecture proposed by Chen and Ru in [1] on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X1,…,Xm) be a non-degenerate decomposable form with coefficients in k. We show that for every finite set of places S of k containing the archimedean places of k, for each real number λ < 1 and each constant c > 0, the inequality [Formula: see text] has only finitely many [Formula: see text]-non-proportional solutions, where HS(x1,…,xm) = Πυ∈S max 1≤i≤m ||xi||υ is the S-height.

A numerical solution for a soft line elastohydrodynamic lubrication contact problem with sinusoidal roughness using the Chebyshev polynomials

2000 ◽  
Vol 214 (5) ◽  
pp. 711-718 ◽  
Author(s):  
M J Jaffar
Keyword(s):  
Chebyshev Polynomials ◽  
Surface Deformation ◽  
Linear Equations ◽  
Elastohydrodynamic Lubrication ◽  
Roughness Model ◽  
Low Elastic Modulus ◽  
Finite Set ◽  
Roughness Amplitude ◽  
Non Linear Equations ◽  
Good Agreement

A sinusoidal roughness model for the elastohydrodynamic problem of a low elastic modulus semi-infinite solid is investigated numerically. Both pressure and surface deformation are expanded in terms of the Chebyshev polynomials of the first kind Tn (x), and hence the governing equations are reduced to a finite set of non-linear equations using the orthogonality property of Tn (x). The influences of the elasticity parameter and the rough surface parameters, such as roughness amplitude, wavelength and phase angle, are examined. The obtained results show good agreement with existing results for a smooth model, and similar characteristics to those reported for a rough elastic layer.

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