Results 1 - 10 of 61

The Weil height in terms of an auxiliary polynomial

by Charles L. Samuels - Acta Arith
"... Abstract. Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number α under certain assumptions on α. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number. ..."
Abstract - Cited by 3 (2 self) - Add to MetaCart
Abstract. Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number α under certain assumptions on α. We prove a theorem which introduces an auxiliary polynomial for giving lower bounds on the height of any algebraic number

A Banach space determined by the Weil height

by Daniel Allcock, Jeffrey, D. Vaaler - Acta Arith
"... Abstract. The absolute logarithmic Weil height is well defined on the quotient group Q × /Tor ` Q ×´ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrical ..."
Abstract - Cited by 7 (1 self) - Add to MetaCart
Abstract. The absolute logarithmic Weil height is well defined on the quotient group Q × /Tor ` Q ×´ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space

Algebraic Numbers of Small Weil’s height in CM-fields: on a Theorem of Schinzel ∗

by Francesco Amoroso , 2009
"... Let K be a CM-field. A. Schinzel proved ([Sch 1973]) that the Weil height of non-zero algebraic numbers in K is bounded from below by an absolute constant C outside the set of algebraic numbers such that |α | = 1 (since ..."
Abstract - Cited by 5 (1 self) - Add to MetaCart
Let K be a CM-field. A. Schinzel proved ([Sch 1973]) that the Weil height of non-zero algebraic numbers in K is bounded from below by an absolute constant C outside the set of algebraic numbers such that |α | = 1 (since

ESTIMATING HEIGHTS USING AUXILIARY FUNCTIONS

by Charles L. Samuels
"... Abstract. Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving the Weil height. The height appears as ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
Abstract. Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving the Weil height. The height appears

The Height in Terms of the Normalizer of a Stabilizer

by John Matthew Garza , 2008
"... This dissertation is about the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We investigate connections between the normalizer of a stabilizer and lower bounds for the Weil height of algebraic numbers. In the archimedean case we extend a result of Schinzel [ ..."
Abstract - Add to MetaCart
This dissertation is about the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We investigate connections between the normalizer of a stabilizer and lower bounds for the Weil height of algebraic numbers. In the archimedean case we extend a result of Schinzel

Points of bounded height on del Pezzo surfaces

by Yu. I. Manin, Yu. Tschinkel - Compositio Math , 1993
"... 0.1. Heights. In this paper, we prove some results on the asymptotic behaviour of the number of algebraic points of bounded height on del Pezzo (and more general) rational surfaces. The basic (Weil) height on a coordinatized projective space over an algebraic number field k is given by the formula ..."
Abstract - Cited by 8 (2 self) - Add to MetaCart
0.1. Heights. In this paper, we prove some results on the asymptotic behaviour of the number of algebraic points of bounded height on del Pezzo (and more general) rational surfaces. The basic (Weil) height on a coordinatized projective space over an algebraic number field k is given by the formula

RIGID LATTICES ARE MORDELL-WEIL

by Michael Larsen , 2004
"... Abstract. We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is Mordell-Weil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one copy ..."
Abstract - Add to MetaCart
Abstract. We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is Mordell-Weil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one

bounds on the projective heights of algebraic points

by Charles L. Samuels - Acta Arith
"... Abstract. If α1,..., αr are algebraic numbers such that r ∑ r∑ N = i=1 αi ̸= i=1 α −1 i for some integer N, then a theorem of Beukers and Zagier [1] gives the best possible lower bound on r∑ log h(αi) i=1 where h denotes the Weil Height. We will extend this result to allow N to be any totally real a ..."
Abstract - Cited by 4 (1 self) - Add to MetaCart
Abstract. If α1,..., αr are algebraic numbers such that r ∑ r∑ N = i=1 αi ̸= i=1 α −1 i for some integer N, then a theorem of Beukers and Zagier [1] gives the best possible lower bound on r∑ log h(αi) i=1 where h denotes the Weil Height. We will extend this result to allow N to be any totally real

Heights On Elliptic Curves And The Diophantine Equation

by Grigor Grigorov, Jordan Rizov , 1998
"... . In this paper we give sharp explicit estimates for the difference of the Weil height and the N'eron -- Tate height on the elliptic curve v 2 = u 3 \Gamma cu. We then apply this in the proof of the fact that if c ? 2 is a fourth power free integer and the rank of v 2 = u 3 \Gamma cu is ..."
Abstract - Add to MetaCart
. In this paper we give sharp explicit estimates for the difference of the Weil height and the N'eron -- Tate height on the elliptic curve v 2 = u 3 \Gamma cu. We then apply this in the proof of the fact that if c ? 2 is a fourth power free integer and the rank of v 2 = u 3 \Gamma cu

Integral points of fixed degree and bounded height. preprint

by Martin Widmer
"... In this article we count algebraic points of bounded Weil height with integral coor-dinates, generating an extension of given degree over a fixed number field k. We derive a precise asymptotic formula for their number as the height gets large. Various related results have appeared in the literature. ..."
Abstract - Cited by 3 (2 self) - Add to MetaCart
In this article we count algebraic points of bounded Weil height with integral coor-dinates, generating an extension of given degree over a fixed number field k. We derive a precise asymptotic formula for their number as the height gets large. Various related results have appeared in the literature
Results 1 - 10 of 61