On polynomial matrix spectral factorization by symmetric extraction

We revise the 1963 Davis algorithm [2] for the spectral factorization of a para-Hermitian nonnegative polynomial matrix
\Phi
, by symmetric factor extraction: this algorithm is careless about zeros at infinity. By introducing the notion of diagonal reducedness of
\Phi
, we obtain an easy sufficient test for the absence of zeros at infinity. We show then how to get
\Phi
, diagonally reduced by diagonal excess reduction steps (similar to the Oono and Yasuura steps), removing all zeros at infinity, and then how to remove synunetrically finite zeros while keeping el, diagonally reduced (hence, free of zeros at infinity). This results in a revised symmetric extraction spectral factorization algorithm with monotone degree control. An example shows the didactical conceptual simplicity of the method. Appropriate symmetric extraction is discovered by revising and discovering important particular one-sided factor extraction properties of polynomial matrices.