Forgetting the obvious (cutting the thing down) what is the better way to kill a palm tree: pouring concentrated poison at its roots, or drilling holes in the tree midway up its trunk and pouring the poison in those holes? One of those ridiculous questions that aarise from ridiculous discussions.
By a bottle of Ortho Palm-Be-Gone see what the instructions say.
What kind of palm, date or coconut? How quickly do you want it to be dead? There was a virus(?) that was slowly creeping its way through the coconut palms of Belize when I was there in 1995. It was easy to notice the ill health of the trees compared to my earlier visit in 1993. I would guess that many of them have died by now. If you don’t mind waiting 6-8 years, the palm blight may be the answer you are looking for.
If you wait awhile, it’ll probably migrate.
IMHO, the quickest way would be to saw the thing down and run it through a big ole honking chipper.
Total time from happy live tree to itsy bitsy chips (depending, of course, on size): 1 to 3 hours.
Of the two methods you originally proposed: I’d go with application of herbicide to the rootzone. You’ll get better uptake than the drill and fill method.
Of course, if the tree’s relatively small, you could spray the canopy with herbicide. That’d be quicker than the root application. Palm death in 3 to 10 days, depending on the herbicide and weather conditions.
I’d still go with the clip and chip method myself . . . where’s the Stihl toting lumberjack smiley when you need it?