Two results on homogeneous Hessian nilpotent polynomials

Abstract

Let z = ( z 1 , … , z n ) and Δ = ∑ i = 1 n ∂ 2 ∂ z i 2 , the Laplace operator. A formal power series P ( z ) is said to be Hessian Nilpotent (HN) if its Hessian matrix Hes P ( z ) = ( ∂ 2 P ∂ z i ∂ z j ) is nilpotent. In recent developments in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (8) (2005) 2201–2205. [MR2138860]; G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (6) (2006) 503–510. [MR2170971]. See also math-ph/0308035; W. Zhao, Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc. 359 (2007) 249–274. [MR2247890]. See also math.CV/0409534], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture (VC) of HN polynomials: for any homogeneous HN polynomial P ( z ) ( of degree d = 4 ), we have Δ m P m + 1 ( z ) = 0 for any m ≫ 0 . In this paper, we first show that the VC holds for any homogeneous HN polynomial P ( z ) provided that the projective subvarieties Z P and Z σ 2 of C P n − 1 determined by the principal ideals generated by P ( z ) and σ 2 ( z ) ≔ ∑ i = 1 n z i 2 , respectively, intersect only at regular points of Z P . Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F = z − ∇ P with P ( z ) HN if F has no non-zero fixed point w ∈ C n with ∑ i = 1 n w i 2 = 0 . Secondly, we show that the VC holds for a HN formal power series P ( z ) if and only if, for any polynomial f ( z ) , Δ m ( f ( z ) P ( z ) m ) = 0 when m ≫ 0 .