# A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix

Given a square, nonsingular matrix of univariate polynomials F∈K[x]^n × n over a field K, we give a fast, deterministic algorithm for finding the Hermite normal form of F with complexity O^∼(n^ωd) where d is the degree of F. Here soft-O notation is Big-O with log factors removed and ω is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal entries of its Hermite normal form, having as cost O^∼(n^ωs) operations with s the average of the column degrees of F.

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