I think the entire post was created by an AI, and I had a similar negative reaction to it. Nevertheless, it's an excellent post with clear explanations of mathematical concepts. This just goes to show how impossible it is to tell the difference between something created by a human and something created by an LLM.
one thing I didn't understand. If the conjecture is false for real numbers, why isn't it false for complex numbers too? Why can't you just take a real number counterexample and use it for complex numbers?
I guess the answer is that a polynomial map whose Jacobian determinant is constant non-zero over the reals may not also be constant non-zero over all the complex numbers.
conjecture said "if there is no J(point) = 0, there's no f(point)=f(point)"
counterexample for reals moved point=0 outside reals, but kept point=point inside
you can see the "jacobian is sum of squares" being mentioned - that is sufficient to say there's no negative values in the reals, but doesn't work for the whole complex field
for complex numbers you have to have jacobian be a constant, or you'll get the zero somewhere
Also more and more use of an extremely annoying effect: text-heavy animations that reset themselves for no reason while you're trying to read them. Step 11, for instance: no value whatsoever is added by erasing and redrawing that figure.
Still, overall a very nice piece of pedagogy. If you didn't grok determinants before, this will probably help.