Combinatorics and Graph Theory

   

Some Facts on Permanents in Finite Characteristics

Authors: Anna Knezevic, Greg Cohen, Marina Domanskaya

The permanent’s polynomial-time computability over fields of characteristic 3 for k-semiunitary matrices (i.e. n×n-matrices A such that ��������(����^�� − ��_��) = ��) in the case 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)×(n-k)sub-permanents (or permanent-minors) of a unitary n×n-matrix and their possible relations, because an (n-k)×(n-k)-submatrix of a unitary n×n-matrix is generically a ksemi-unitary (n-k)×(n-k)-matrix. The following paper offers a way to receive a variety of such equations of different sorts, in the meantime extending (in its second chapter divided into subchapters) this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for a generic matrix’s sub-permanents 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 demonstrates quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3. Besides, the paper’s 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 square matrix in characteristic 5 and conjecturing the existence of a similar scheme in characteristic 3. Throughout the paper, we investigate various matrix compressions and transformations preserving the permanent and related functions in certain finite characteristics. And, as an auxiliary algebraic tool supposed for an application when needed in all the constructions we’re going to discuss in the present article, we’ll introduce and utilize a special principle involving a field’s extension by a formal infinitesimal and allowing, provided a number of conditions are fulfilled, to reduce the computation of a polynomial over a field to solving a system of algebraic equations in polynomial time

Comments: 85 pages; this research was partly supported by the School of Electrical Engineering, Computing and Mathematical Sciences of the Curtin University (Australia) whose member is one of the authors (Anna Knezevic)

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Submission history

[v1] 2019-08-16 08:15:44
[v2] 2019-08-30 02:58:42
[v3] 2019-08-30 08:45:46

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