One potential solution is for Amazon to do a better job of encouraging users to rate every product they buy. That would quickly drown out the fake reviews and make such tactics uneconomical.

Off the top of my head I would suggest that on the post-checkout screen they should list the customer's most recent purchases asking "Please rate your current satisfaction with your recent purchases". Just a quick 5-star rating, a small button to write a textual review if the user wants, and a note clearly stating that their review can be changed later if their opinion changes.

I suggest putting that at post-checkout because it's a point where Amazon still has the user engaged, so asking is not annoying (like their emails currently are), but the user is done shopping so they aren't preoccupied with something else. In other words, it is the least annoying time to ask. They should, of course, also have a rating box on the home page (again, a list of recent purchases, possibly muxing in older purchases at random).

And then weighting reviews is super important. Reviews given by users later in their ownership of the product should be weighted higher than reviews given just after receiving the product. Changed reviews should be weighted higher, as well as reviews on returns.