A colleague suggested to look at the Popov form. I didn't look at what Sage 
is currently doing, so my apologies if this turns out to not be a useful 
comment.

Here is a random paper on this that I found [1].

Bill.

[1] http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/issac96.pdf

On Tuesday, 15 November 2016 10:01:09 UTC+1, Kwankyu Lee wrote:
>
> Hi,
>
> The current Hermite normal form algorithm in Sage for matrices over k[x] 
> seems embarrassingly slow. It is a generic algorithm for matrices over 
> PIDs. What would be possible ways to improve the situation? Any reference 
> to literature or implementations or like would be welcome. Thanks.
>
>

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