Solution of the congruence problem for arbitrary Hermitian and skew-Hermitian matrices over polynomial rings (Q1565800)

The paper presents a reply to a problem raised by \textit{V. G. Kac} (private communication, October 2001). It is shown in agreement with the statement by V. G. Kac that every Hermitian or skew-Hermitian matrix over the complex polynomial algebra \(\mathbb C[t]\) is congruent to the direct sum of \(1\times 1\) matrices and \(2\times 2\) matrices with zero diagonal. It is also proved that if two \(n\times n\) Hermitian or skew-Hermitian matrices have the same invariant factors, then they are congruent. Other fields are also considered, assuming that the characteristic is not 2.