A Polynomial-time Algorithm for Computing the Permanent in GF(3^q)

A polynomial-time algorithm for computing the permanent in any field of characteristic 3 is presented in this article. The principal objects utilized for that purpose are the Cauchy and Vandermonde matrices, the discriminant function and their generalizations of various types. Classic theorems on the permanent such as the Binet-Minc identity and Borchadt's formula are widely applied, while a special new technique involving the notion of limit re-defined for fields of finite characteristics and corresponding computational methods was developed in order to deal with a number of polynomial-time reductions. All the constructions preserve a strictly algebraic nature ignoring the structure of the basic field, while applying its infinite extensions for calculating limits. A natural corollary of the polynomial-time computability of the permanent in a field of a characteristic different from 2 is the non-uniform equality between the complexity classes P and NP what is equivalent to RP=NP( Ref. [1]). Unless specified otherwise, all the results are for fields of Characteristic 3. Definitions and denotations. For m n -matrices m n h ij def h m n ij ij def a A b a B A B A        } { , } { , ,  ,                q q h h def h h A A A ... 1 1 ) ,..., ( (for real numbers q h h h ,..., , 1 ) (the Hadamard product, degree and vector-degree), for ) , ( } ,..., 1 { }, ,..., 1 { J I A m J n I   is the sub-matrix of A whose set of rows is I and of columns J , for matrices ] [ ] 1 [ ,..., s A A with the same number of rows  s r r A 1 ] [  denotes the matrix   ] [ ] 1 [ | ... | s A A ,      n i ij m j a A scal 1 1 ) ( ,   ; s) for vector (as ) ( ' ), ( simply write will we 1 ) dim( for , ) ( ... ) ( ' , ) matrix e Vandermond the ( ) ( ) ( , ) ( ] [ ] [ ] [ ) dim( 1 ] [ )] [dim( ) 1 ,..., 0 ( ] [ t van t van t t Van t d d t d d t Van t Van t Van t t Van h h h t def h t def h T def h