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B oolean circuits of polynomial size and poly-logarithmic depth are givenf or computing the Hermite and Smith normal forms of polynomial matrices overfinite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficient sequential algorithms. Furthermore, we give a polynomial-time deterministic sequential algorithm for the Smith normal form overthe rationals. The Smith normal form algorithms are applied todoi:10.1137/0608057 fatcat:wi5nzwifljfifpxvshky2cd6pm