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The polynomial-time computability of the permanent over fields ofcharacteristic 3 for k-semi-unitary matrices (i.e. square matrices such thatthe differences of their Gram matrices and the corresponding identity matricesare 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 thecomputational complexity boundaries for the permanent modulo 3. Now we extendthis 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 andtheir possible relations, because an (n-k)x(n-k)-submatrix of a unitarynxn-matrix is generically a k-semi-unitary (n-k)x(n-k)-matrix. The followingpaper 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 setof polynomial-time permanent-preserving reductions and equations for thesub-permanents of a generic matrix they might yield, including a number ofgeneralizations and formulae (valid in an arbitrary prime characteristic)analogical to the classical identities relating the minors of a matrix and itsinverse. Moreover, the second chapter also deals with the Hamiltonian cyclepolynomial in characteristic 2 that surprisingly possesses quite a number ofproperties very similar to the corresponding ones of the permanent incharacteristic 3, while over the field GF(2) it obtains even more amazingfeatures. Besides, the third chapter is devoted to the computational complexityissues of the permanent and some related functions on a variety of Cauchymatrices and their certain generalizations, including constructing apolynomial-time algorithm (based on them) for the permanent of an arbitrarymatrix in characteristic 5 (implying RP = NP) and conjecturing the existence ofa similar scheme in characteristic 3.

Some facts on Permanents in Finite Characteristics