Computation of the Maximal Degree of the Inverse of a Cubic Automorphism of the Affine Plane with Jacobian 1 via Gröbner Bases

If k is any commutative ring, k[X,Y ] will denote the algebra of polynomials with coefficients in k in the indeterminates X,Y and Ak = Spec k[X,Y ] the affine plane over k. A kendomorphism f of Ak will be identified with its coordinate functions f = (f1, f2), where fi (i = 1, 2) belongs to k[X,Y ]. We define the Jacobian of f by Jac(f) = ∂f1 ∂X ∂f2 ∂Y − ∂f1 ∂Y ∂f2 ∂X and the degree of f by deg(f) = max1≤i≤2 deg(fi). Let d be a non-negative integer and f an endomorphism of AC whose degree is less than or equal to d. The Jacobian Conjecture in degree d(CJ(d)) states that f is invertible if and only if its Jacobian is a non-zero constant. Let Cd be the smallest integer C such that if k is a Q-algebra and f a k-automorphism of Ak satisfying Jac(f) = 1 and deg(f) ≤ d, then we have deg(f−1) ≤ C. Bass has proven the following result in Bass (1983):