A simple question with a collection of remarkably different answers.
I’m going back to my original statement, with a small tweak from econ prof’s answer:
Step 1 – tell your neighbor the problem and come to a solution if you can. Personally, I’d try to convince them to remove the tree at their expense and see what they say. Show them the damage and let them know that in lieu of cutting the roots, you thought you would ask nicely.
If they whine about it, go to step 2 which is file a claim with your insurance company citing the damage done and just let them go to work. Make sure you tell the neighbor that you will be doing this before you do it. I know if my neighbor came to me with this problem, I would be OK with it – I would want to see what my insurance co said about it.
Either your insurance co will tell you you are hosed; they will tell you to chop the roots as you wish;they will pay for the tree to be removed; or they will get in touch with the neighbor’s insurance co and hash out a solution. Either way, they are going to know the rights.