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By Arne Storjohann

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This chapter gives algorithms to compute each of these forms. 1: Non canonical echelon forms over a PIR include the time to recover a unimodular transform matrix U ∈ Rn×m such that U A = H. Some of our effort is devoted to expelling some or all of the logorithmic factors in the cost estimates in case a complete transform matrix is not required. 1 shows how to transform A to echelon or minimal echelon form. 2 shows how to transform an echelon form to satisfy also (r2). 3 simply combines the results of the previous two sections and gives algorithms for the Hermite and minimal Hermite form.

Proof. Compute a modified Gauss transform (F, R, ∗, ∗, ∗) of AT and let T = F RAT . Let D be the diagonal matrices with same diagonal entries as T . Then ∗1 L ∗2 A =(D−1 T )T ((D−1 F D)T )−1 ((DR)T )−1 . 5 The Triangularizing Adjoint Let A ∈ Rn×n , R an integral domain. It is a classical result that the determinant of A can be computed in O(nθ ) ring operations. Here we show that all leading minors of A can be recovered in the same time. Recall that the adjoint of A is the matrix Aadj with entry Aadj ij equal i+j to (−1) times the determinant of the submatrix obtained from A by deleting row j and column i.

Let R be a PIR. Every A ∈ Rn×m is left equivalent to an H ∈ Rn×m that satisfies: (r1) Let r be the number of nonzero rows of H. Then the first r rows 53 54 CHAPTER 3. TRIANGULARIZATON OVER RINGS of H are nonzero. For 0 ≤ i ≤ r let H[i, ji ] be the first nonzero entry in row i. Then 0 = j0 < j1 < j2 < . . < jr . (r2) H[i, ji ] ∈ A(R) and H[k, ji ] ∈ R(R, H[i, ji ]) for 1 ≤ k < i ≤ r. (r3) H has a minimal number of nonzero rows. Using these conditions we can distinguish between four forms as in Table 3.

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