  The polynomial-time computability of the permanent over fields of
characteristic 3 for k-semi-unitary matrices (i.e. square matrices such that
the differences of their Gram matrices and the corresponding identity matrices
are of rank k) in the case k = 0 or k = 1 and its #3P-completeness for any k >
1 (Ref. 9) is a result that essentially widens our understanding of the
computational complexity boundaries for the permanent modulo 3. Now we extend
this result to study more closely the case k > 1 regarding the
(n-k)x(n-k)-sub-permanents (or permanent-minors) of a unitary nxn-matrix and
their possible relations, because an (n-k)x(n-k)-submatrix of a unitary
nxn-matrix is generically a k-semi-unitary (n-k)x(n-k)-matrix. The following
paper offers a way to receive a variety of such equations of different sorts,
in the meantime extending this direction of research to reviewing all the set
of polynomial-time permanent-preserving reductions and equations for the
sub-permanents of a generic matrix they might yield, including a number of
generalizations and formulae (valid in an arbitrary prime characteristic)
analogical to the classical identities relating the minors of a matrix and its
inverse. Moreover, the second chapter also deals with the Hamiltonian cycle
polynomial in characteristic 2 that surprisingly possesses quite a number of
properties very similar to the corresponding ones of the permanent in
characteristic 3, while over the field GF(2) it obtains even more amazing
features. Besides, the third chapter is devoted to the computational complexity
issues of the permanent and some related functions on a variety of Cauchy
matrices and their certain generalizations, including constructing a
polynomial-time algorithm (based on them) for the permanent of an arbitrary
matrix in characteristic 5 (implying RP = NP) and conjecturing the existence of
a similar scheme in characteristic 3.
