On vectors whose span contains a given linear subspace
Let \(k>\ell>m>0\) be integers. The main aim is to estimate \[ c(k,\ell,m)=\sup\inf H(S)^{(\ell-k)/(k-m)}\prod_{i=1}^\ell|\mathbf p_i|, \] where the supremum is taken over all subspaces \(S\) of \(\mathbb Q^k\) of dimension \(m\), and the infimum is taken over all sets of linearly independent vectors \(\mathbf p_1,\dots,\mathbf p_\ell\) in \(\mathbb Z^k\), whose span contains \(S\). Here \(H(S)\) is the determinant of the lattice \(S\cap\mathbb Z^k\) and \(|\mathbf p|\) is the Euclidean norm of \(\mathbf p\). A related problem is the decomposition of integer vectors, introduced by the second author [Bull. Pol. Acad. Sci. Math. 35, 155--159 (1987; Zbl 0639.10023)], and investigated in a series of papers by the first two authors, S. Chaładus and Yu. Teterin.
scientific article; zbMATH DE number 2162483