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

for a scalar  , m n   is the m n  -matrix all whose entries equal  , for

) dim(a -vector whose coordinates are pairs of non-decreasing nonnegative integer finite sequences, including the empty one. The rows and columns containing the term i a will be called the i aextension (and i a itself the extension’s value ); ) , (

we’ll call the extension-degree of i a , where i  is the left extension-degree, i  is the right one, | | i  is the extension’s height , | | i  is its width , and

then we’ll say that the matrix has an i a -singularity. Singularities with (0,0)  i  will be called simple. The sum

will be called the total balance of the extension-degrees’ vector  .

The formal sum

will be defined as the extensionplane of i a , ) ( i i weight  will also be called the weight plane -of i  in i  , the pair ) , ( i i a  will be called an extension-variety and thus we can also talk about the E-sum of a on the extension-planes (i. e. sub-vectors of the vector-variety) will be called right and left prolongations (correspondingly).

We will denote the E-sum of a on  by ) , (  a esum (we’ll use the brackets   only for left and right extension-degrees whose cardinalities exceed 1);

for an r n  -matrix A and an

is even :

the E-generated functions (for vectors with appropriate dimensions):

the star-function:

We’ll call the extension-plane (0,0) the wave and  the biwave; together with their values (say i z and j w ) such extension-varieties will be called a i z -wave and a j w -biwave (correspondingly) ; the wave-function (a partial case of the 2-waves-function):

For both the wave-function and the 2-waves-function i  will be called the i -th ebb, and  the ebb vector. the base-function:

if F is a basic field and  is a formal variable then for

we’ll denote the Jacobian matrix

Preliminaries.

The Binet-Minc identity (for any characteristic, Ref.

Let A be an

The Binet-Minc identity for Characteristic 3: .

For our purposes we’ll also use this identity in the form

The generalized Binet-Minc identity for ) , ( B A coper ).

Proof:

the right part can be re-written as

Then, by the construction,

means the set of sets with removed  -s.

The generalized Binet-Minc identity for ).

Proof:

by analogy with the previous proof, we re-write the right part as

Then, by the construction,

In the generic case this method allows to reduce A to a triangular matrix when computing ) det(A .

Lemma 3 (for any characteristic):

The neighbouring computation principle (for any characteristic):

exists and is nonsingular.

Then, given  , over

where:

This method will be called the ng neighbouri computation of the function

is computable in polynomial time for any h in the region

and there exists a bearing point, then ) ( f is computable in polynomial time for any  too.

In the further, to be clear and short, we’ll call a system of functions S algebraically absolutely independent in a region R (given by a system of equations with a zero right part) iff the joint system of functions consisting of S and the left part of the system representing R is algebraically independent at some point of R.

a left (right) prolongation pattern of a vector-variety can be removed with an induced coordinate-wise transformation of the remaining vector of extension-planes such that the E-sum preserves its value. Such a transformation will be called a left (right) prolongation-derivative, or the prolongation-derivative on the given prolongation. Formally it will be denoted (correspondingly for the left and right cases) by

This principle is based on the Binet-Minc identity.

Let A be an n n  -matrix, B be a

The Borchardt formula (for any characteristic, Ref. [2]) :

The main part.

Theorem I :

Then Proof:

)

is the matrix of the coefficients of the polynomials in the formal scalar variable ) , ( ),…, , (

. The determinant of this matrix is a homogenous polynomial in

, and in the meantime it is divided by

; moreover, its coefficients are 1 and -1, therefore ))

, what completes the proof.

and, basing on Theorem I, we calculate the limit to get the lemma’s equality.

and then, basing on Corollary I.1, we again just calculate the limit.

Proof: basing on Lemma 2, we conclude that the right part of this equality is

It follows from the definition of the copermanent and Theorems I, III.

• —————————————————————————————Let’s note that for 0 ) , ( "

for any n pair-wise distinct values of  (due to the characteristic),

for instance for all the coordinates of x , i.e. it’s equivalent to

Theorem V :

Then there exist scalar constants 3 2 ,   such that the system of functions in

let us consider the Jacobian matrix which