Solution of the congruence problem for arbitrary hermitian and skew-hermitian matrices over polynomial rings

Let ${}^*$ be the involutorial automorphism of the complex polynomial algebra ${\mbox{\bf C}}[t]$ which sends $t$ to $-t$. Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to the direct sum of $1\times1$ matrices and $2\times2$ matrices with zero diagonal. Moreover we show that if two $n\times n$ hermitian or skew-hermitian matrices have the same invariant factors, then they are congruent. The complex field can be replaced by any algebraically closed field of characteristic $\ne2$.

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Published 1 January 2003