I'm having a major problem with tree roots cracking my tarmac drive.

I'm not sure exactly where the roots are coming from, I have a beech hedge
alongside the driveway, about 8 yeras old and only 4 ft high. Immediately
behind that is a Leylandii hedge that my neighbour planted about 10 years
ago which he keeps trimmed about 8 ft high - I'm much more suspicious of the
leylandii.

I don't want really want to cut down my own beech hedge - I planted it in
front of the leylandii because I detest leylandii hedging - and I don't want
to ask my neighbour to kill off his hedge when I'm not certain that is
casuing the problem.

What I was thinking of doing was to drill some holes in the tarmac where the
cracks are and try to kill off the roots at that point, keeping my fingers
crossed that the hedge will survive with the roots going in the opposite
direction.

Is this likely to work and can anyone recommend a suitable chemical to
apply?