Jacobian conjecture in $\mathbb R^2$

Abstract

Jacobian conjecture states that ifis a polynomial map such that the Jacobian of F is a nonzero constant, then F is injective.This conjecture is still open for all n ≥ 2, and for both C n and R n . Here we provide a positive answer to the Jacobian conjecture in R 2 via the tools from the theory of dynamical systems.Let (f, g) : R 2 (C 2 ) → R 2 (C 2 ) be a polynomial map. We denote by J(f, g) the Jacobian matrix of the map (f, g), and by D(f, g) the Jacobian of (f, g),The classical Jacobian conjecture states that if the Jacobian D(f, g) = 1, then F is injective. This conjecture was first posed as a question by Keller [33] in 1939. For more information on the history of this conjecture, see e.g. the survey paper [6,23,53]. Nowadays, the Jacobian conjecture is formulated in the next form (see e.g. Bass et al [6] and Essen [23, pages XV and 82]).Associated to the Jacobian conjecture, Randall [41] in 1983 posed the so called real Jacobian conjecture.Real Jacobian conjecture. If F : R 2 → R 2 is a polynomial map such that the Jacobian DF of F does not vanish, then F is injective. This conjecture is not correct in general, as illustrated by Pinchuck [39] in 1994, which we will recall it again later on.A general formulation in C n of the real Jacobian conjecture was posed by Smale [46] in 1998 as his 16th problem on a list of 18 open mathematical problems. For distinguishing it from the Jacobian conjecture mentioned above and according to Pinchuck [39], we call it Strong Jacobian conjecture.Strong Jacobian conjecture. If F : C n → C n is a polynomial map such that the Jacobian DF of F does not vanish, then F is injective.

Injectivity of non-singular planar maps with one convex component

“…For the special case of planar polynomial maps with constant, non-zero Jacobian determinant, the level set connectedness was considered in [21,22,23]. In [22,23] an approach was proposed based on the commutativity of the Hamiltonian flows having P (x, y) and Q(x, y) as Hamiltonian functions, similarly to what was done in [19].…”

“…The key point in the proof of our main result is the level set connectedness. This property has already been used in previous papers [21,22,23].…”