When I was digging out* a patch to grow veggies earlier in the spring I found big roots in the soil unconnected to anything in our garden. It looks like they're coming from some kind of palm trees planted right at the edge of our neighbour's yard. They were so thick I couldn't break them with the spade so they're just lying there underneath the patch for now.
Now this spot where the neighbouring roots are is the spot that's best suited (with regard to sunlight and airflow) to growing veggies and herbs. Can I just claim the soil for our plants and chop away the neighbouring roots or would that be bad form? What would you do and why?
Now, if the neighbour's tree was something nice like a lemon tree I wouldn't even think of doing this, which suggests that the answer is not straightforward at all.
Note: We have not met this particular neighbour because their yard backs on to ours rather than being beside ours. Obviously part of the answer might be to talk to that neighbour, but I am looking for a gardening-informed answer rather than the political one!
(*) Our soil is very poor in every way, so this is a process I go through square metre by square metre adding compost, mulch and a little organic fertiliser in order to grow anything at all.
UPDATE: Adding pictures (click to enlarge) so you can see how precariously close not just the vege patch but a nectarine tree I planted at the end of winter and a few years old citrus are to the neighbouring palms (which by the way are towering at approx 5m+ high).