The Jacobian Conjecture: linear triangularization for homogeneous polynomial maps in dimension three

Let k be a field of characteristic zero and F : k → k a polynomial map of the form F = x + H, where H is homogeneous of degree d ≥ 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T−1HT = (0, h2(x1), h3(x1, x2)), where the hi are homogeneous of degree d. As a consequence of this result, we show that all generalized Drużkowski mappings F = x+H = (x1 +L d 1, . . . , xn +L d n), where Li are linear forms for all i and d ≥ 2, are linearly triangularizable if JH is nilpotent and rk JH ≤ 3. Introduction The Jacobian Conjecture asserts that every polynomial map F : Cn → Cn satisfying the Jacobian hypothesis, i.e. det JF ∈ C∗ is invertible. It was shown in [1] and [14] that it suffices to prove the Jacobian Conjecture for all polynomial maps of the form F = x+H, where H = (H1, . . . ,Hn) and each Hi is a homogeneous polynomial of some fixed degree d (which we may assume to be 3). For such F the Jacobian hypothesis det JF ∈ C∗ is well-known to be equivalent to the nilpotency of the matrix JH ([1] or [6]). Therefore one is naturally led to the study of nilpotent Jacobians. A fundamental open problem in this respect is the following, which was formulated as a conjecture problem by various authors ([5], [6], [8], [9], [10]). Homogeneous Dependence Problem HDP (n). Let H = (H1, . . . ,Hn) : k n → kn be homogeneous of degree d ≥ 2 such that JH is nilpotent. Are the rows of JH linearly dependent over k or equivalently are the Hi linearly dependent over k (k is a field of characteristic zero). Affirmative answers are known in the following cases: rk JH ≤ 1 (also if H is not homogeneous), [1], [6]. In particular, this holds for the case n = 2. The case n = 3 and d = 3 (Wright [13], 1993) and n = 4, d = 3 (Hubbers, [8], 1994, see also [6]). One of the main results of this paper (Theorem 1.2) gives an affirmative answer for n = 3 (d arbitrary). As a consequence we will show that in dimension 3 the Jacobian ∗Supported by the Netherlands Organisation of Scientific Research(NWO)