I know in most municipalities that if tree branches are 'hanging' over your side of the property line, you are allowed to cut/trim them, up to the property line. As for what you can do about the root system is an answer I don't have. Assuming you're gonna ask your neighbor to have the tree removed and they say 'NO', the question is do you have the right to cut/trim the roots on YOUR side of the property line. With this logic, I'm gonna bet that if you DID have the legal right to cut/trim the root system on your side of the line, you'll more than likely kill your neighbor's tree and create the Hatfields vs the McCoys. Even if you offered to pay the neighbor to have the tree removed (or even asked if you could do it yourself), they may still say no and what can you do about the roots?!
First, I would think the answer to your specific question is a broad one. In a worst case scenario where your neighbor says NO, NO, and NO to what you want to do, you'll need to find out if you have any rights. First, I'd check with your township / town / city about your rights. If you don't get an answer to you, pay a lawyer $75 to get the legal answer. $75 is not that much $$$ when you have to realize that chances are good you'll be living next to this neighbor for some time to come, and will have to deal with any 'fallout' from whatever actions you take that are NOT in line with your neighbors'. You may want to talk to your insurance agent too to see if your filter AND pool are BOTH covered from tree root damage. If they say no, you have more ammunition to present to your neighbor as to the consequences of in-action (wether it's the neighbor's inaction or your inaction). Also, after seeing the lawyer, if need be, you can also present to the neighbor your legal remedies (assuming you have any).
If I was your neighbor, I'd have some respect for you in that at least you took the time to investigate the insurance and legal side-of-things and would, thus, understand why you would do 'what you must'. As your neighbor, assuming you have the right to cut the roots on your side of the line, I may not like it but it's your right and 'I' was given the opportunity to act. One last thing ... assuming you can cut the roots, you know you may kill the tree. Tell this to the neighbor because it may be better and more cost-effective to simply have the tree removed and the stump ground-out. Before you do anything to the tree or roots (assuming you can do 'something' and since I don't know anything about this type of tree) you may want to find out from a Nursery if cutting the tree down and grinding the roots are sufficient to kill the root system, thus, solving your problem. It would be a waste of $$$ if cutting down the tree and grinding the stump DON'T kill the root system!!! My best to ya and hope this helps. I'd be real interested in hearing back from you on what your legal rights are, what your insurance company's position is, and what your neighbor's decision is with you on this.