You have a right to remove the roots, but you must act reasonably. If the removal of the roots causes the tree to die, you could be held responsible. I might consult with an arborist first to determine if the root removal can be done safely. In any event, the damage being caused by your neighbor's tree is his responsibility and constitutes a continuing nuisance and a trespass. You have a right to go to him and demand that he address the situation by removing the tree or trimming the roots on his property as well as to ask him to compensate you for your damages. Note, however, that the statute of limitations is 3 years, so you would only have the right to recover damages occurring within the last three years. In that this is a continuing situation, however, with damages continuing to accrue, you have a continuing right to pursue him for those damages if he does not address the problem.
Attorney Quinn is spot on.....I would only add that if you have suffered $4k in damage you should make a claim against your homeowner insurance. They will make a claim against him/his homeowner's insurance, and you should be able to be compensated for your loss.
Attorney Quinn and Ferris are correct and provide strong advice. I would only add the following - be careful if you pursue the litigation route with a neighbor. Many of these cases end up costing more in legal fees then the amount of damage done to your property. If you haven't already, perhaps have an informal conversation with your neighbor and see if this issue can be resolved between the two of you.
There is potential liability if you cut the roots and the tree dies. While the tree roots encroaching under the land constitute a nuisance as set forth in Civil Code 3479, there is a 1994 appellate court case (Booska v. Patel) in which the court held that the "basic policy of this state .. is that everyone is responsible for any injury casued another by his want of ordinary care or skill in the management of his property." Thus, while historical decisions seemingly gave the right to cut limbs, roots, branches at the property line, the more modern approach is to balance a landlowner's right to control his/her own property with a duty to act reasonably.