Two Results on Homogeneous Hessian Nilpotent Polynomials

Abstract

Let $z=(z_1,...,z_n)$ and $\Delta ={\sum }_{i=1}^n\frac{{\partial }^2}{\partial z_i^2}$ the Laplace operator. A formal power series $P(z)$ is said to be {\it Hessian Nilpotent}(HN) if its Hessian matrix $\Hes P(z)=(\frac {\partial^2 P}{\partial z_i\partial z_j})$ is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have ${\Delta }^m{P}^{m+1}(z)=0$ 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 ${\mathcal{Z}}_P$ and ${\mathcal{Z}}_{{\sigma }_2}$ of $\mathbb{C}{P}^{n-1}$ determined by the principal ideals generated by $P(z)$ and ${\sigma }_2(z):={\sum }_{i=1}^n{z}_i^2$, respectively, intersect only at regular points of ${\mathcal{Z}}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z-\nabla P$ with $P(z)$ HN if $F$ has no non-zero fixed point $w\in {\mathbb{C}}^n$ with ${\sum }_{i=1}^n{w}_i^2=0$. Secondly, we show that the VC holds for a HN formal power series $P(z)$ if and only if, for any polynomial $f(z)$, ${\Delta }^m(f(z)P(z{)}^m)=0$ when $m>>0$.

Latex, 7 pages

Cited by in corpus (2)

• A Vanishing Conjecture on Differential Operators with Constant Coefficients
• Some Properties of and Open Problems on Hessian Nilpotent Polynomials