Why do we want residue to break down anyway? Some say we want to release the nutrients. That may be especially true in the case of nitrogen. More likely we don't want to tie up the nitrogen with residue breakdown because we want it to be furnishing nutrients for our crop.
No-till Farmer ran this
article on residue breakdown , but it offers little in the "how" department. One of the reasons I see for wanting residue to break down, is for soil warmth and aeration. If we are in a high residue cropping system, if the soil is completely covered, it may keep soil wet in the spring. Properly done vertical tillage can help break up the residue and start decomposition in three ways. One is that we make the pieces smaller. Two is that we mix in small bits of soil which contains microbes that inoculate the residue to decompose. Three is that we expose a bit of soil to allow improved air and water movement. Keep in mind that vertical tillage should be shallow and should not bury residue. Another tool that will have the same effect is the Aerway. If those tools are not available, a very light disking could help. The disk should be in the ground an inch or less.
Some people like a bit of UAN over the top to break up residue. The shortcoming is that it is really a waste of nitrogen. Something else to keep in mind is that the root residue in the soil maybe more of a drag on yields in corn on corn than the residue on top. Perhaps a small amount of nitrogen incorporated would help with that issue. The residue on the surface is not really the problem with nitrogen tie-up, because it has limited contact with the rooting zone.
Good drainage is also important for good residue breakdown. Tile drainage will promote better aeration, but even good surface drainage can help.