Bamboo is the relative who comes to stay for a weekend and never leaves.
A few simple rules.
First, unless prohibited, you can pretty much plant what you want on your property. As plants grow, they do not respect boundaries. If your neighbor's plant grows onto your property, you can cut it off at the boundary, whether below, on or above the ground. But you can't kill the plant and you can't cross the boundary.
Second, bamboo is kind of like a child. They start out little and under your control, but grow up and leave. Once they do, you lose control over them. The bamboo in your neighbor's yard is not yours. It is now the neighbor's, or the neighbor's problem depending on how they look at it. We see a lot more complaints when bamboo crosses the boundary than people who are thankful.
As for keeping the bamboo out, most nurseries can help you with a product that creates a physical barrier underground, but you still have to be vigilant. You can do just about anything you want, but only on your property.
I agree with Mr. Davis' response. The ownership of plants or trees growing on a boundary line is shared by the two property owners. You are entitled to trim and remove vegetation on your property; however you must respect the property interests of your neighbor. You must be careful not to trespass on your neighbor's property including not damaging any vegetation on their property. I suggest working with your neighbor to reach an agreement regarding the bamboo. If both of the parties (you and your neighbor) feel comfortable putting something in writing, an attorney can help you draft a simple agreement.