Power Linear Keller Maps of Dimension Three

In this paper it is proved that a power linear Keller map of dimension three over a field of characteristic zero is linearly triangularizable. Let K be a field. A polynomial map F in dimension n over K is an n-tuple (F1, F2, · · · , Fn) of polynomials in K[X1, X2, · · · , Xn]. If G is another polynomial map of the same dimension, then the composition of F and G is defined by F ◦G = (F1(G1, G2, · · · , Gn), · · · , Fn(G1, G2, · · · , Gn)). The polynomial map F is invertible if there exists a polynomial map G such that F ◦G and G◦F are both identities. It is Keller if the determinant of its Jacobian is a nonzero element in K, i.e., det JF ∈ K∗. By the chain rule for Jacobians, invertible polynomial maps are Keller maps. The famous Jacobian conjecture states that if charK = 0, then any Keller map is invertible (see, e.g., [1] or [4]). A polynomial map F is power linear if it is of the form (X1 +A1 1 , X2 +A d2 2 , · · · , Xn + An n ) where Ai is a linear form in X1, X2, · · · , Xn and di ≥ 2 for all i. It is cubic linear if it is power linear where di = 3 for all i. Druzkowski [2] showed that in the case charK = 0 if cubic linear Keller maps are invertible, then the Jacobian conjecture would be true. A polynomial map is triangular if it is of the form (X1 +p1, X2 +p2, · · · , Xn+pn) where pi is a polynomial in K[Xi+1, Xi+2, · · · , Xn]. It is linearly triangularizable if there exists a linear invertible polynomial map φ such that φ ◦ F ◦ φ−1 is triangular. A polynomial map is elementary if it is of the form (X1, · · · , Xi−1, Xi + p,Xi+1, · · · , Xn) where p ∈ K[X1, · · · , Xi−1, X̂i, Xi+1, · · · , Xn]. It is tame if it can be written as a composition of invertible linear maps and elementary maps. It is not hard to see that linearly triangularizable maps are tame and tame maps are invertible. The tame generators conjecture asserts that an invertible polynomial map is tame. This is proved in dimension two by Jung [5] and van der Kulk [6]. For dimensions beyond two, it is an open problem. In this note we prove that power linear Keller maps in dimension three over a field of characteristic zero are linearly triangularizable (hence tame and invertible) proving both the Jacobian and the tame generators conjecture in this case. (It is worth noting that if the degrees of the components of these maps are all equal to three, then the result is a special case of that of Wright [7] which states that all cubic homogeneous polynomial maps are linearly triangularizable.) Received by the editors August 1, 1999 and, in revised form, February 2, 2000. 2000 Mathematics Subject Classification. Primary 14R15, 14R10.