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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
SCG - Symbolic Computation Group
3 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
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 $\widetilde{\mathcal{O}}(n^\omega \lceil s \rceil)$ 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.
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George Labahn, Vincent Neiger, Wei Zhou. Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix. Journal of Complexity, Elsevier, 2017, ⟨10.1016/j.jco.2017.03.003⟩. ⟨hal-01345627v2⟩

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