Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Abstract
Given a nonsingular
n×n
matrix of univariate polynomials over a field
K
, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use
O
˜
(
n
ω
⌈s⌉)
operations in
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
ω
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
O
˜
(
n
ω
)
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.