Hessian Nilpotent Polynomials and the Jacobian Conjecture

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Let z=(z₁,..., zₙ) and Δ=∑ᵢ₌₁ⁿ ² z²ᵢ the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to the following what we call vanishing conjecture: for any homogeneous polynomial P(z) of degree d=4, if Δᵐ Pᵐ(z)=0 for all m ≥ 1, then Δᵐ Pᵐ⁺¹(z)=0 when m>>0, or equivalently, Δᵐ Pᵐ⁺¹(z)=0 when m> 32 (3ⁿ⁻²-1). It is also shown in this paper that the condition Δᵐ Pᵐ(z)=0 (m ≥ 1) above is equivalent to the condition that P(z) is Hessian nilpotent, i.e. the Hessian matrix P(z)=(² P zᵢ zⱼ) is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen BE1 and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.