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

Given a nonsingular n x 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) over tilde (n(omega) inverted right perpendicular s inverted left perpendicular) 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 omega 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) over tilde (n(omega)) 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. (C) 2017 Elsevier Inc. All rights reserved.