depending how big the roots are it can take several years for roots to decompose, first the stump and roots will die but takes a long time for shoots to rot, you really need to get the stump out and cut any offending shoots out, as its in lawn not sure you can use anything such as root out as it may kill your lawn to ;o((

I recently bought a spray gun of Roundup. The good thing is that it comes out as a frothy mixture so not only does it stick to the offending item but it doesn't go all over the place.