Two results on homogeneous Hessian nilpotent polynomials

Let z=(z1,…,zn) and image, the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent (HN) if its Hessian matrix image 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 ΔmPm+1(z)=0 for any mmuch greater-than0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z) provided that the projective subvarieties image and image of image determined by the principal ideals generated by P(z) and image, respectively, intersect only at regular points of image. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F=z−backward differenceP with P(z) HN if F has no non-zero fixed point image with image. Secondly, we show that the VC holds for a HN formal power series P(z) if and only if, for any polynomial f(z), Δm(f(z)P(z)m)=0 when mmuch greater-than0.