What you described in your first message seemed similar to the approach used in the degree N root solving algorithm by Cem Yuksel; splitting the curve in simpler segments, then bisect into them. I'd be happy to explore what you suggested, but I'm not mathematically literate, so I'll be honest with you; what you're saying here is complete gibberish to me, and it's very hard to follow your point. It will take me weeks to figure out your suggestion and make a call as to whether it's actually simpler or more performant than what is proposed in the article.

I have written some gist to illustrate the approach I suggest. The code run but there may be bugs, and it don't use the appropriate optimizer. The purpose is to illustrate the optimisation approach.

https://gist.github.com/unrealwill/1ad0e50e8505fd191b617903b...

Point 33 "intersection between bezier curve with a circle" may be useful to find the feasible regions of the subproblems https://pomax.github.io/bezierinfo/#introduction

The approach I suggest will need more work, and there are probably problematic edge cases to consider and numerical stability issues. Proper proofs have not been done. It's typically some high work-low reward situation.

It's mostly interesting because it highlight the link between roots and local mimimum. And because it respect the structure of the problem more.

To find roots we can find a first root then divide the polynomial by (x-root). And find a root again.

If you are not mathematically literate, it'll probably be hard to do the details necessary to make it performant. But if you use a standard black-box optimizer with constraints it should be able to do it in few iterations.

You can simplify the problem by considering piece-wise segments instead of splines. The extension to chains of segment is roughly the same, and the spatial acceleration structure based on branch-and-bound are easier.