Cut off the sprouts. Apply/paint a stump/brush killer to the cuts. You can get a poison made for that kind of application at the Big Box.

Ha! I once posted a diatribe about how many YEARS it took for all the root shoots to cease after I had two huge Paulownia removed from a newly purchased property.

They were relentless.

I cut them off at the ground nearly every morning the first season post removal and there were hundreds. I kept cutting to starve them out and they kept coming. That process went on a couple of years then I noticed them slowing down a bit- I thought.

Only to see sprouts coming up in a neighbor yard across the street.

I didn't consider them eradicated for at least five years.

And now that it's over the enormous root mass is decaying underground and I have sunk spots.

In most residential lots these trees are a PLAGUE!

Normally I use concentrate triclopyr on fresh cut stumps to kill things, but the sheer numbers of these things would make that extremely time consuming.

Depends on how badly you want them gone and how quickly.