Minimal bases of three-dimensional complete lattices (Q2759301)

The authors follow the classical ideas of Voronoi, Berwick and Minkowski, developed by \textit{H. Appelgate} and \textit{H. Onishi} [J. Number Theory 15, 283--294 (1982; Zbl 0504.12012)] and \textit{J. Buchmann} [ibid. 26, 8--30 (1987; Zbl 0615.12001)]. Let \(\mathcal L_3^*(\mathbb R)\) denote the set consisting of the three-dimensional lattices \(\Gamma\) in \(\mathbb R^3\) such that the components of every non-zero vector in \(\Gamma\) are all non-zero. For any non-empty finite subset \(S\subset\mathbb R^3\), let \(|S|_i\) denote the maximum of the absolute values of the \(i\)th components of the vectors in \(S\). For \(\Gamma\in\mathcal L_3^*(\mathbb R)\), a set \(S=\{\gamma^{(1)},\dots,\gamma^{(t)}\} \subset\Gamma\setminus\{0\}\) is called minimal if \(\gamma^{(i)}\neq \pm\gamma^{(j)}\) for all \(1\leq i< j\leq t\) and there is no \(\gamma\in \Gamma\setminus\{0\}\) such that \(|\gamma_i|<|S|_i\) for \(i=1,2,3\). If this holds for \(t=1\), \(\gamma^{(1)}\) is a called a minimum of \(\Gamma\). If the conditions hold for \(t=2\), then \(\gamma^{(1)}, \gamma^{(2)}\) are called neighbours or neighbouring minima. A basis \(\gamma^{(1)},\gamma^{(2)},\gamma^{(3)}\) of a lattice \(\Gamma\in \mathcal L_3^*(\mathbb R)\) is called a minimal basis if \(\{\gamma^{(1)},\gamma^{(2)},\gamma^{(3)}\}\) is a minimal set. The following results are proved:NEWLINENEWLINENEWLINE1. If \(\gamma^{(1)},\gamma^{(2)},\gamma^{(3)}\) is a basis of \(\Gamma\in \mathcal L_3^*(\mathbb R)\) and the matrix with these column vectors can be transformed by compositions of permutations and changes of signs into one of two special forms, then it is a minimal basis.NEWLINENEWLINENEWLINE2. If \(\{\gamma^{(1)},\gamma^{(2)},\gamma^{(3)}\}\) is a minimal set and the vectors \(\gamma^{(i)}\) are linearly independent, then \(\gamma^{(1)},\gamma^{(2)},\gamma^{(3)}\) is a basis (a different proof was already given by Minkowski), and the matrix can be transformed as above.NEWLINENEWLINENEWLINE3. Every pair \(\gamma^{(1)},\gamma^{(2)}\) of neighbouring minima can be complemented by a third point \(\gamma^{(3)}\) up to a minimal basis.