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
Comments
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