Post #3968823

@gregeganSF@mathstodon.xyz - Wow, that's the most amazing thing I've heard for a while! I've never heard any good reason to think this conjecture is true, but I assumed people had looked for counterexamples so hard that any counterexample would have to be much more complicated than this one. "The Jacobian conjecture proposed that if you have a map F:C^n→C^n with polynomials as the components, and the determinant of the matrix of partial derivatives of F is a non-zero constant, then F will have a polynomial inverse." I find that this statement, on Wikipedia, sort of weakens the blow. Instead of saying "is a nonzero constant" I would say "everywhere nonzero", since that makes the conjecture sound more general. It's equivalent, but it sounds more impressive. Talking to myself, I'd actually say "if F: ℂⁿ→ℂⁿ is a polynomial map whose Jacobian is invertible everywhere, F has a polynomial inverse." That clarifies the charm of the conjecture: it's an insane relative of the inverse function theorem, which says that a smooth function whose linearization at a point is invertible must have a smooth inverse in some neighborhood. We change "smooth" to "polynomial", "at a point" to "at every point", and "in some neighborhood" to "everywhere". Puzzle: Find a smooth F: ℝⁿ→ℝⁿ whose Jacobian is invertible at each point but where F does not have a smooth inverse.