Some facts on Permanents in Finite Characteristics
Abstract
The polynomialtime computability of the permanent over fields of characteristic 3 for ksemiunitary 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 #3Pcompleteness 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 (nk)x(nk)subpermanents (or permanentminors) of a unitary nxnmatrix and their possible relations, because an (nk)x(nk)submatrix of a unitary nxnmatrix is generically a ksemiunitary (nk)x(nk)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 polynomialtime permanentpreserving reductions and equations for the subpermanents 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 polynomialtime 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.01783
 Bibcode:
 2017arXiv171001783K
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 89 pages