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