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So if "there are no good, general methods for solving systems of more than one nonlinear equation," then what hope do we have in this matter of Bezier intersection? The key is to recognize that we need a good, specific (as opposed to general) method, specific to the Bezier equations. Indeed, there is another algorithm altogether, which avoids the inherent numerical confusion of bisection, and which allows us to find the peaks, no more and no less. I am not going to reveal that unpublished algorithm here, but I will give you a hint: one avoids the mire of a swamp by reformulating it as a large set of problems that have a small number of fast, closed-form solutions.