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

Abstract

Let * be the involutorial automorphism of the complex polynomial algebra 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 11 matrices and 22 matrices with zero diagonal. Moreover we show that if two n 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 = 2.