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

George Labahn (1) , Vincent Neiger (1, 2, 3) , Wei Zhou (1)

1 Symbolic Computation Group
2 ENS de Lyon - École normale supérieure de Lyon
3 ARIC - Arithmetic and Computing

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 $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ 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 $\omega$ 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 $\widetilde{\mathcal{O}}(n^\omega)$ 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.

Mots clés: Hermite normal form; determinant; polynomial matrix

Domaines: Calcul formel [cs.SC]

George Labahn, Vincent Neiger, Wei Zhou. Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix. Journal of Complexity, 2017, ⟨10.1016/j.jco.2017.03.003⟩. ⟨hal-01345627v2⟩