ANNALES POLONICI MATHEMATICI LXXVI.1-2 (2001) The Real Jacobian Conjecture for polynomials of degree 3 by Janusz Gwoździewicz (Kielce) Abstract. We show that every local polynomial diffeomorphism (f,g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism. 1. Introduction. In [4] Pinchuk presented a polynomial mapping F : R 2 → R 2 such that F is not a global diffeomorphism although Jac(F ) > 0 everywhere in R 2 . Components of Pinchuk’s mapping have degrees 10 and 35. It is an interesting question what is the lowest degree of a polynomial map in an example like this. In this note we prove that it should be at least 4. Theorem 1. Every polynomial mapping (f,g): R 2 → R 2 with a positive Jacobian such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism. Recall that Jac(f,g) is given by Jac(f,g)= f ′ x g ′ y − f ′ y g ′ x . The condition Jac(f,g) > 0 guarantees that (f,g) is a local diffeomorphism. For the proof of our main result we need a sequence of lemmas. 2. Lemmas Lemma 1. Let (f,g): R 2 → R 2 be a polynomial mapping with a positive Jacobian. If for all t ∈ R the level sets {f = t} are connected then (f,g) is a global diffeomorphism. Proof. Every injective polynomial mapping from R 2 to itself is bijective (see [1], [5]). Therefore it suffices to show that (f,g) is an injection. Suppose to the contrary that (f,g)(p 1 )=(f,g)(p 2 )=(t, s) for p 1 = p 2 . Let T be a segment of a curve {f = t} joining points p 1 and p 2 . Take another point p 3 ∈ T such that g(p 3 ) = max p∈T g(p) or g(p 3 ) = min p∈T g(p). From Lagrange’s multipliers method it follows that the derivatives df (p 3 ) and 2000 Mathematics Subject Classification : Primary 14P99. Key words and phrases : Jacobian conjecture, Newton polygon, real polynomial map- ping. [121]