# Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix

Given a nonsingular n × n matrix of univariate polynomials over a field K, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use O(n^ω s ) operations in K, where s is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and ω is the exponent of matrix multiplication. The soft-O notation indicates that logarithmic factors in the big-O are omitted while the ceiling function indicates that the cost is O(n^ω) when s = o(1). Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.

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