Siegel lemmas of avoidment (Q2893037)

Let \(n\geq 1\) be an integer and \(\Omega\) a subset of the space \(\mathbb R^n\) endowed with a given norm \(\|\cdot\|\). The successive minima \(\lambda_1,\dots,\lambda_n\) of \((\Omega,\|\cdot\|)\) are defined as follows: for \(1\leq i\leq n\), \(\lambda_i\) is the minimum of the numbers \(r>0\) such that there exist \(\omega_1,\dots,\omega_i\) in \(\Omega\), linearly independent over \(\mathbb R\), of norms \(\leq r\). Minkowski geometry of numbers deals with the case where \(\Omega\) is a lattice in \(\mathbb R^n\). A special case of the result of the text under review involves the complement \(\Omega\setminus Z\) of an algebraic closed subset \(Z\) in a lattice \(\Omega\), in which case the authors obtain an upper bound for \(\lambda_1(\Omega\setminus Z, \|\cdot\|)\) in terms of \(\lambda_s(\Omega, \|\cdot\|)\) for some \(s\geq 1\).NEWLINENEWLINEMore generally, they consider an adelic vector bundle in place of \((\Omega, \|\cdot\|)\). In this context, upper bounds for the minima \(\lambda_i\) are called Siegel's Lemmas, while the word ``évitement'' (avoidment) refers to the complement of a closed algebraic subset in the vector bundle. This topic has already been investigated by a few authors only, namely \textit{L. Fukshansky} [Monatsh. Math. 147, No. 1, 25-41 (2006; Zbl 1091.11024); J. Number Theory 120, No. 1, 13--25 (2006; Zbl 1192.11018); J. Number Theory 130, No. 10, 2099--2118 (2010; Zbl 1282.11073)] and the second author [Manuscr. Math. 130, No. 2, 159--182 (2009; Zbl 1231.11076)].NEWLINENEWLINEThe main results of the present paper relate the minima of the complement of an algebraic subset to the minima of the adelic vector bundle. One of the results is an ``absolute'' one, meaning that the dependence on the number field disappears.