Your going to have to keep mowing and chopping sprouts. Round up applied while sprouts are activly growing, but Round up will also kill other plants including grass. A broad leafed herbicide applied while shoots are activly growing will not harm grass but will other nroad leaf plants. You might have to do this for years, the roots are all interconnected, and as long as one shoot is alive it keeps roots alive.
The sumps are connected to roots too and are acting as source of food for roots. To encourage stump rot, drill large holes 1" or greater from top, as many and as deep as you can. Drill close to bark but do not let hole come out thru bark. There are products sold to promote stump rot, one is called "Stump Rot" pour it in holes.
but many home concoctions work just as well or better. Heap compost or aged manure on top. My Grandmother had a concoction that kills stump, promotes rot and may get into root system to start killing it. Fill holes with epsom salts/ regular salt mixture, then slowly pour vegetable oil into holes until they won't hold any more. Especially holes near outer side where live wood is. After that oil is absorbed pour turpentine into holes til they won't take any more. It starts killing but and oils in it, being from trees, carry salts thru out tree as does vegetable oil. Pine oil can be used not a cleaner with pine oil but stronger, also linseed oil, raw is best boiled will work. Just about any vegetable oil. Kerosene, which she called "coal oil" can be used but doesn't disperse through tree as readily Repeat the treatments as oils and salts disappear. The turps kill stump fast, you mite want use just oils first hoping salts get into root system B4 stump dies cutting off flow.
The shoots you just have to keep cutting. 10% vinegar sprayd on shoots will burn them up and any other plant but leaves no residue, other plants will eventually grow back, will take more than one application as it does not kill roots. Eventually when shoots are gone roots will no longer have food to share or keep selves alive.
