The Jacobian Conjecture

Jacobian conjecture: let \(F: \mathbb{C}^n \to \mathbb{C}^n\) be a polynomial map whose Jacobian determinant is a nonzero constant. Then \(F\) is invertible with polynomial inverse.

Keller asked it in 1939 for two variables; the general case grew into one of the central open problems of affine algebraic geometry, advertised on Smale's 1998 list. The two-variable case is settled (Jung 1942, van der Kulk 1953); the three-variable case is the subject of its own active entry here. Reductions show it suffices to treat cubic homogeneous maps (Bass–Connell–Wright 1982; Drużkowski 1983), and Tsuchimoto (2005) and Belov-Kanel–Kontsevich (2007) tied its fate to Dixmier's conjecture — yet no proof or counterexample has emerged.