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

A Canonical Form for Positive Definite Matrices

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension $n$, but it is practical and more efficient than canonical forms based on Minkowski reduction

Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pad\'e approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ field operations, where $\omega$ is the exponent of matrix multiplication and the notation $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pad\'e approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta(m^2 \sigma)$, the cost bound $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $\mathcal{O}(m \sigma)$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms