Small points on subvarieties of a torus
type: Item
instance of: scholarly article
title: Small points on subvarieties of a torus
zbMATH Open document ID: 1234.11081
DOI: 10.1215/00127094-2009-056
author: Francesco Amoroso
author: Evelina Viada
published in: Duke Mathematical Journal
publication date: 2010-03-30T00:00:00Z
full work available at URL: 20100126133824 4b5ee240c8237.pdf
review text: The authors improve earlier estimates by \textit{W. M. Schmidt} [Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 235, 157--187 (1996; Zbl 0917.11023)], \textit{S. David} and \textit{P. Philippon} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, No.3, 489--543 (1999; Zbl 1002.11055); Errata ibid. 29, 729--731 (2000)], and \textit{F. Amoroso} and \textit{S. David} [Compos. Math. 142, No. 3, 551--562 (2006; Zbl 1116.11045)] on lower bounds for the heights of points on subvarieties of linear tori. The most involved tools used by the authors are upper and lower bounds for the Hilbert function, due to Chardin and Philippon. Apart from that, the authors' proof is much simpler than the arguments of David and Philippon and Amoroso and David. The following notation is used. We view \({\mathbb G}_{{\mathbf m}}^n\) as a subset of \({\mathbb P}^n\) by identifying \((x_1,\ldots ,x_n)\) with \((1:x_1:\cdots :x_n)\). Thus, a subvariety of \({\mathbb G}_{{\mathbf m}}^n\) is the intersection of a subvariety of \({\mathbb P}^n\) with \({\mathbb G}_{{\mathbf m}}^n\). The height \(h\) on \({\mathbb G}_{{\mathbf m}}^n(\overline{{\mathbb Q}})\) is the absolute logarithmic Weil height from \({\mathbb P}^n({{\mathbb Q}})\). Given a subset \(S\) of \({\mathbb G}_{{\mathbf m}}^n(\overline{{\mathbb Q}})\), we denote by \(S(\varepsilon )\) the set of points \({\mathbf x}\in S\) with \(h({\mathbf x})\leq\varepsilon \) and by \(\overline{S(\varepsilon )}\) the Zariski closure in \({\mathbb G}_{{\mathbf m}}^n\) of this set. Given a subvariety \(V\) of \({\mathbb G}_{{\mathbf m}}^n\), we denote by \(\delta (V)\) the minimum of all integers \(\delta\) such that \(V\) is the intersection of hypersurfaces of degree at most \(\delta\). Further, we denote by \(\delta_0(V)\) the minimum of all integers \(\delta\) with the property that there exists an intersection \(X\) of hypersurfaces of degree at most \(\delta\) such that each irreducible component of \(V\) is an irreducible…
reviewed by: Jan-Hendrik Evertse
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 5688238
zbMATH Keywords: Heights
zbMATH Keywords: subvarieties of tori
zbMATH Keywords: Bogomolov's conjecture
zbMATH Keywords: small points
MaRDI profile type: Publication
OpenAlex ID: W2103130565
cites work: Solving algebraic equations in roots of unity.
cites work: MINORATION DE LA HAUTEUR NORMALISE DANS UN TORE
cites work: Points de petite hauteur sur une sous-variété d'un tore
cites work: Q4937710
cites work: Q4853852
cites work: Une majoration de la fonction de Hilbert et ses conséquences pour l'interpolation algébrique
cites work: Q4699502
cites work: Q4937826
cites work: Q4428502
cites work: Linear equations in variables which lie in a multiplicative group
cites work: A Bogomolov property for curves modulo algebraic subgroups
cites work: Autour d'une conjecture de Serge Lang. (Around a conjecture by Serge Lang)
cites work: Exponential diophantine equations
cites work: Lemmes de zéros dans les groupes algébriques commutatifs
cites work: On alternative heights. III
cites work: Mordell-Lang plus Bogomolov
cites work: Q4797991
cites work: Approximate formulas for some functions of prime numbers
cites work: Q4233884
cites work: The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve
description: scientific article; zbMATH DE number 5688238