Two Results on Homogeneous Hessian Nilpotent Polynomials

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Let z=(z₁,..., zₙ) and Δ=∑ᵢ₌₁ⁿ ∂²/∂ z²ᵢ the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent(HN) if its Hessian matrix P(z)=(∂² P/∂ zᵢ∂ zⱼ) is nilpotent. In recent developments in [BE1], [M] and [Z], 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 Δᵐ Pᵐ⁺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 Zₚ and Zσ₂ of C Pⁿ⁻¹ determined by the principal ideals generated by P(z) and σ₂(z):=∑ᵢ₌₁ⁿ zᵢ², respectively, intersect only at regular points of Zₚ. 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∈ Cⁿ with ∑ᵢ₌₁ⁿ wᵢ²=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), Δᵐ (f(z)P(z)ᵐ)=0 when m>>0.