A note on Siegel's lemma over number fields
type: Item
instance of: scholarly article
title: A note on Siegel's lemma over number fields
zbMATH Open document ID: 0839.11011
DOI: 10.1007/BF01294863
author: Damien Roy
author: Jeffrey Lin Thunder
published in: Monatshefte für Mathematik
publication date: 1996-06-19T00:00:00Z
full work available at URL: https://eudml.org/doc/178708
review text: Let \(K\) be a number field of degree \(d\) and discriminant \(D_K\). Define \(\Delta_K = \sqrt {|D_K |}\). The height \(H(a)\) of a vector \({\mathbf a} \in K^n\) is the projectively invariant absolute Weil height. Define \(H'(a)\) in the same way, but with the Euclidean norm at archimedean places instead of the sup norm. The inhomogeneous height \(H_1 ({\mathbf a})\) is \(H(1, {\mathbf a})\), while the height \(H'(A)\) of a matrix \(A\) with rows \({\mathbf a}_1, \dots, {\mathbf a}_M\) is \(H'( {\mathbf a}_1 \wedge \cdots \wedge {\mathbf a}_M)\). Let \(S\) be a system of \(M\) linearly independent homogeneous linear forms in \(N\) variables with coefficients in \(K\). The height \(H'(S)\) of the system \(S\) is the height \(H'(A)\) of its coefficient matrix \(A\). Assume \(N > M\). \textit{E. Bombieri} and \textit{J. Vaaler} have shown [Invent. Math. 73, 11-32 (1983; Zbl 0533.10030)] that there exists a basis \({\mathbf x}_1, \dots, {\mathbf x}_{N - M} \) of the solution space of \(S\) in \(K^N\) such that \[ \prod^{N - M}_{i = 1} H ({\mathbf x}_i) \leq \bigl( (2/ \pi)^{rsb2} \Delta_K \bigr)^{(N - M)/d} H'(S). \] A similar estimate holds (with the inhomogeneous height \(H_1)\) for a basis of the solution space of \(S\) in the ring of integers \({\mathcal O}_K\) of \(K\). The authors show that these estimates are sharp. For \(d,N,M\) positive integers with \(d > 1\) and \(N \geq M + 2\), there exists a constant \(C_1 > 0\) depending only on \(d\) and \(N - M\) such that, for any number field \(K\) of degree \(d\), there exists a system \(S\) of \(M\) linearly independent homogeneous linear forms in \(N\) variables with coefficients in \(K\) for which any basis \({\mathbf x}_1, \dots, {\mathbf x}_{N - M}\) of the solution space in \(K^N\) satisfies \[ \prod^{N - M}_{j = 1} H ({\mathbf x}_i) \geq X\quad\text{where}\quad X = C_1 \Delta_K^{\bigl[ (N - M - 1)/2 \bigr] /d(d - 1)} H'(S). \] As pointed out by D. W. Masser to the authors (private communication), their resul…
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11D04
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11H16
zbMATH DE Number: 838725
zbMATH Keywords: Siegel's lemma
zbMATH Keywords: number fields
zbMATH Keywords: Weil height
zbMATH Keywords: homogeneous linear forms
zbMATH Keywords: inhomogeneous height
zbMATH Keywords: discriminant
MaRDI profile type: Publication
Wikidata QID: Q125057471
cites work: On Siegel's lemma
cites work: Q4739993
cites work: On heights of algebraic subspaces and diophantine approximations
cites work: Lower bounds for height functions
cites work: Q3139250
description: scientific article; zbMATH DE number 838725