Vashti wrote:
I've just taken over an allotment which is horsetail from head to toe,
and I'm currently trimming it daily or every other day. I know I have to
remove any root piece or fertile brown stalks that I find. My question is,

You will never gain control of horsetails without use of chemical
weapons. Glyphosate applied after bruising the stems is about as good
as anything. You need a translocating weedkiller to damage the root
system to maximum effect as they go down forever.

This is one case where regular application of under strength weedkiller
(unlicensed use) applied regularly to the weed after bruising or
strimming high works better than the at label strength. The plant has a
silica exoskeleton and is almost waterproof unless bruised first.

can the green ferny stems root themselves and keep growing, or can I safely
leave them where they are when they've been cut down?

I have never known the top growth reroot, but the pieces of the root do
all too easily

Once the soil is in good cultivation you can usually pull them out with
a decent length of root as they appear (quite satisfying) - a never
ending task if there is anywhere else nearby they can grow unmolested.
Mine came in from a railway enbankement.

You can't beat it but you can keep it in check.

Regards,
Martin Brown