The Jacobian conjecture is one of my favorite unsolved conjectures, because it's so simple to state (C = the complexes): Suppose F : C^2 —> C^2 is a polynomial map F(z, w) = (P(z,w), Q(z,w)) meaning: P and Q are each a polynomial over C. Then: If* we assume that F is locally one-to-one, then F is a) globally one-to-one and b) onto. I believe the truth status of neither a) nor b) is known at present. Amazing! Here's another related question I just saw in Terry Tao's math blog: ----- Does there exist a *polynomial bijection* P : Q x Q —> Q where P is a polynomial in two variables with rational coefficients i.e., P is a member of Q[X, Y]. ----- He says he thinks this very unlikely, but says that a proof (if true) is likely to be very hard to come by. So, it's evidently an open problem! Obviously a *necessary* condition is that P be locally one-to-one Q x Q —> Q. Which is just saying that for all (x,y) in Q^2, we have dP/dx * dP/dY ≠ 0 (i.e., not both derivatives are zero). So the obvious more general questions: Given *any* field K (or given any subfield K of Q^ = the algebraic closure of the rationals): ----- Does there exist a bijection F : K x K —> K, where F(X,Y) is a polynomial with coefficients in K ??? ----- [[[[[ Puzzle (tricky!): Find an entire function f(z) — i.e., f(z) is analytic on all of C — that is locally one-to-one everywhere without being globally one-to-one ... or else prove that such cannot exist. ]]]]] —Dan ————— * "Locally one-to-one" means, for each point pt = (z,w) of C^2, there is a neighborhood U of pt = (z,w) in C^2 (C^2 is topologically R^4) such that if for any pt_1, pt_2 both in U we have F(pt_1) = F(pt_2), then pt_1 = pt_1.