Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con