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

Abstract

Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\widetilde{\mathcal{O}}(n^ω\lceil s \rceil)$ operations in $\mathbb{K}$, 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 $\widetilde{\mathcal{O}}(n^ω)$ 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.

34 pages, 3 algorithms