Some results on the parametrization of complex Hadamard matrices

In this paper we provide an analytical procedure which leads to a system of (n − 2)2 polynomial equations whose solutions give the parametrization of the complex n × n Hadamard matrices. It is shown that in general the Hadamard matrices depend on a number of arbitrary phases and a lower bound for this number is given. The moduli equations define interesting geometrical objects whose study will shed light on the parametrization of the Hadamard matrices, as well as on some interesting geometrical varieties defined by them.