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

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.Comment: 34 pages, 3 algorithm

Inversion of Toeplitz Matrices with Only Two Standard Equations

It is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations. The inverse matrix is represented by two of its columns (which are the solutions of the two standard equations), and the entries of the original Toeplitz matrix

Inversion of Toeplitz matrices with only two standard equations

AbstractIt is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations. The inverse matrix is represented by two of its columns (which are the solutions of the two standard equations) and the entries of the original Toeplitz matrix