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This thesis gives a deterministic algorithm to transform a row reduced matrix to canon-
ical Popov form. Given as input a row reduced matrix R over K[x], K a ﬁeld, our algorithm
computes the Popov form in about the same time as required to multiply together over
K[x] two matrices of the same dimension and degree as R. Randomization can be used to
extend the algorithm for rectangular input matrices of full row rank. Thus we give a Las
Vegas algorithm that computes the Popov decomposition of matrices of full row rank. We also show that the problem of transforming a row reduced matrix to Popov form is at least
as hard as polynomial matrix multiplication.