Computing Canonical Bases of Modules of Univariate Relations

We study the computation of canonical bases of sets of univariate relations $(p_1,\dots ,p_m)\in \mathbb{K}[x{]}^m$ such that $p_1 f_1 + \cdots + p_m f_m = 0$;here,theinputelements$f_1,\ldots,f_m$arefromaquotient$$\mathbb{K}$[x]^n/$\mathcal{M}$$,where$$\mathcal{M}$$isa$$\mathbb{K}$[x]$-module of rank $n$ given by a basis $\mathbf{M}\in \mathbb{K}[x{]}^{n\times n}$ in Hermite form. We exploit the triangular shape of $\mathbf{M}$ to generalize a divide-and-conquer approach which originates from fast minimal approximant basis algorithms. Besides recent techniques for this approach, we rely on high-order lifting to perform fast modular products of polynomial matrices of the form $\mathbf{P}\mathbf{F}\mathbf{M}$. Our algorithm uses $O\stackrel{~}{}(m^{\omega -1}D+n^{\omega }D\mathrm{/}m)$ operations in $\mathbb{K}$, where $D=\mathrm{d}\mathrm{e}\mathrm{g}(\det (\mathbf{M}))$ is the $\mathbb{K}$-vector space dimension of $\mathbb{K}[x{]}^n\mathrm{/}\mathcal{M}$, $O\stackrel{~}{}(\cdot )$ indicates that logarithmic factors are omitted, and $\omega$ is the exponent of matrix multiplication. This had previously only been achieved for a diagonal matrix $\mathbf{M}$. Furthermore, our algorithm can be used to compute the shifted Popov form of a nonsingular matrix within the same cost bound, up to logarithmic factors, as the previously fastest known algorithm, which is randomized.