D. Det, A)) )/ log(D(det(A))) is resp

, M k+1 , which is a well defined integer matrix. recursively be applied for the resulting chain . . . M k?1 M k M k+1 . . . as long as an admissible entry is found

, Conditions 1 and 2 are trivially fulfilled for modular images of exact chains of matrices over Z. It suffices to take Z = X = 0 and W = Y = 0. Thus, Thm. 16.2.8 can repeatedly be applied in order to compute local smith forms at a prime p for exact chains of integer matrices

, Suppose that M k is an exact chain of matrices over Z and let us take M k mod q. One reduction as in Thm. 16.2.8 is possible i.e. the Smith forms agree after reduction. However, attempting another reduction may fail

, As a related property we also notice, that for matrices over Z p l all minimal generating sets of columns/rows have the same, minimal cardinality equal to the number of non-zero invariant factors modulo p l . Algorithm LRE of [38, 39] can be adapted to prove this claim. Therefore, by finding a dependency, we may remove the dependent vector from the generating set in order to obtain a generating set of minimal cardinality at the end of the process, The proof of Thm. 16.2.8 is based on the special form of zero-divisors in Z p l compared with Z q

, 8 allow to trace bases of the kernel (resp. image) of M k , which become LE k (resp. E k?1 U ), if E k (resp E k?1 ) is the initial base. This is an important factor in some applications, see e.g. for computing homologies of maps

, Over integers, computational issues regarding coefficient swell arise, which is discussed in [64]. However, this can be seen as a good point of the method

, V i?1 , V i , K) repeatedly finds admissible entries and perform reductions according to Thm. 16.2.2, maintaining marks, so that V i?1 and V i contain information about the rows and columns that have to be zeroed in the neighboring matrices, Procedure Reduce(M i+1 , V i ) reads matrix M i?1 from data file and zeroes rows from M i?1 which are marked in V i . P artialElimination(M i

, N , a finite exact chain of matrices, Ensure: D k , k = 1

, sequence of diagonal matrices, Ensure: M k , k =, vol.1

, N , an exact chain of matrices, such that SF (M k ) = SF (diag(D k , M k )), k = 1

K. D-i-=-p-artialelimination(m-i-,-v-i?1-,-v-i,

, Reduce(M i+1, vol.6

V. D-n-=-p-artialelimination(m-n, V. N-?1, and K. ,

K. D-i-=-p-artialelimination(m-t-i-,-v-i-,-v-i?1,

, = P artialElimination(M T 1

, N , an exact sequence of matrices over a PIR R is given at the entrance of Alg. 17.2.1. Let 1 < i ? N . During the course of the algorithm, whenever P artialElimination

, ?1 ) is performed, as matrix M i?1 is changed. According to Thm. 16.2.2 this can be repaired by a deletion of rows of M i, PROOF At the beginning of the algorithm M k is an exact sequence and thus the conditions are fulfilled. The condition M i?1 M i = 0 is first violated when P artialElimination on

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