Two results on homogeneous Hessian nilpotent polynomials

Let z=(z1,…,zn)z=(z1,…,zn) and Δ=∑i=1n∂2∂zi2, the Laplace operator. A formal power series P(z)P(z) is said to be Hessian Nilpotent   (HN) if its Hessian matrix HesP(z)=(∂2P∂zi∂zj) 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)P(z) (of degree  d=4d=4), we have  ΔmPm+1(z)=0ΔmPm+1(z)=0for any  m≫0m≫0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z)P(z) provided that the projective subvarieties ZPZP and Zσ2Zσ2 of CPn−1CPn−1 determined by the principal ideals generated by P(z)P(z) and σ2(z)≔∑i=1nzi2, respectively, intersect only at regular points of ZPZP. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F=z−∇PF=z−∇P with P(z)P(z) HN if FF has no non-zero fixed point w∈Cnw∈Cn with ∑i=1nwi2=0. Secondly, we show that the VC holds for a HN formal power series P(z)P(z) if and only if, for any polynomial f(z)f(z), Δm(f(z)P(z)m)=0Δm(f(z)P(z)m)=0 when m≫0m≫0.