Singular points of cyclic weighted shift matrices. (English) Zbl 1283.15117

Summary: Let \(S\) be an \(n\)-by-\(n\) cyclic weighted shift matrix, and \(F_S(t, x, y) = \det(tI + x \operatorname{Re} (S) + y \operatorname{Im}(S))\) be a ternary form associated with \(S\). We investigate the number of singular points of the curve \(F_S(t, x, y) = 0\), and show that the number of singular points of \(FS(t,x,y)=0\) associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to \(n(n - 3)/2\). Furthermore, we verify that the upper bound \(n(n - 3)/2\) is sharp for \(4 \leqslant n \leqslant 7\).


15B99 Special matrices
14H20 Singularities of curves, local rings
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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