The codimension of degenerate pencils
The paper is devoted to the computation of the codimension of a variety of matrices. If \(d_n\) \([d_n(r)]\) denotes the codimension of the set of pairs of \(n\times n\) Hermitian [real symmetric] matrices \((A,B)\) for which \(\det(\lambda I - A - xB)= p(\lambda,x)\) is a reducible polynomial it is shown that \(d(r)\le n - 1\), \(d_n\le n - 1\) \((n\) odd), \(d_n\le n\) \((n\) even). It is shown that the equality holds for \(n=2,3\), and it is conjectured that equality holds for all \(n\).
scientific article; zbMATH DE number 3765956