Covering numbers under small perturbations
Let \({\mathcal M}([n])\) be a matroid without loops. Let \({\mathcal M}^{(k)}\) be the sum of \(k\) copies of \({\mathcal M}([n]).\) The covering number of \(x, x\in [n],\) is the smallest positive integer \(s_x(M)\) such that \(x\) is a coloop of \({\mathcal M}^{(s)}\); see \textit{J. A. Dias da Silvia, R. Fernandes} and \textit{A. Fonseca} [Linear Algebra Appl. 271, 191-219 (1998; Zbl 0889.05036)]. This notion is particularly interesting if \({\mathcal M}([n])\) is representable (coordinatizable) over a field \(\mathbb{F}\). The author investigates how the covering number of the elements of a representable matroid \({\mathcal M}(S),\) \(S:=\{s_1,\dots, s_n\} \subset \mathbb{F}^d,\) can be changed under arbitrary small perturbations of the vectors \(s_1,\dots, s_n.\)
scientific article; zbMATH DE number 1790260