Serre’s Reduction and the Smith Forms of Multivariate Polynomial Matrices

Algorithm 1

: A matrix equivalence (ME) algorithm.

(i)	Step 1. Declare the ring over which the matrix is defined by declaring the indeterminates and the field of coefficients. Factor the determinant of . Check that the determinant of is the form . If yes, set , the polynomial it corresponds is , , the polynomial it corresponds is , go to Step 2. Otherwise, return this method is not fit for .
(ii)	Step 2. Compute the reduced Gröbner basis of ideal generated by the lower minors of . If , go to Step 3; otherwise, return this method is not fit for .
(iii)	Step 3. Set , , , , and .
(iv)	Step 4. Substitute in to obtain . Compute a ZRP vector such that by using the function SyzygyModule. Then compute a unimodular matrix with is its last column by using the function CompleteMatrix. Compute such that . Compute and set . Compute and obtain , store . , .
(v)	Step 5. When , go to Step 4. When , do Step 6; otherwise, compute , let return .
(vi)	Step 6. Substitute in to obtain , do procedure similar to the step 4. And obtain a ZRP vector such that and a unimodular matrix with is its last column. Then compute such that , and compute and set . Set , substitute in to obtain . Compute a ZRP vector such that by using the function SyzygyModule. Then compute a unimodular matrix with is its last column by using the function CompleteMatrix. Compute and such that . Compute and obtain , where . Store , . Go to Step 5.