Two results on homogeneous Hessian nilpotent polynomials

Let z=(z1,�,zn) and , the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent (HN) if its Hessian matrix 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 ?mPm+1(z)=0 for any m0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z) provided that the projective subvarieties and of determined by the principal ideals generated by P(z) and , respectively, intersect only at regular points of . 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 with . Secondly, we show that the VC holds for a HN formal power series P(z) if and only if, for any polynomial f(z), ?m(f(z)P(z)m)=0 when m0.