zhao-vanishing

Status and infobox

• status: — → refuted
• tier: — → V3
• kind: — → conjecture
• year: — → 2007
• statement: — → Let $\Delta$ be the Laplacian on $\mathbb C[z_1,\dots,z_n]$. If $P$ is homogeneous of degree 4 and $\Delta^mP^m=0$ for all $m\ge1$ (equivalently, $P$ is Hessian nilpotent), then $\Delta^mP^{m+1}=0$ for all large $m$.
• short: — → VC — for Hessian-nilpotent quartic P, Δᵐ Pᵐ⁺¹ = 0 for large m

Relations

• + uses de-bondt-van-den-essen-symmetric: By the symmetric reduction, the Jacobian conjecture reduces to gradient maps $z-\nabla P$ with $P$ a Hessian-nilpotent homogeneous quartic (Zhao's argument starts from the de Bondt–van den Essen symmetric form).
• + uses formal-inverse: So the map has a polynomial inverse exactly when those terms vanish for large m, which is a statement about the formal inverse.
• + equivalent to jc-n: The result is that the vanishing conjecture in every dimension is equivalent to the Jacobian conjecture in every dimension. The equivalence does not preserve the dimension, because it passes through the cubic-homogeneous and symmetric reductions.
• + special case of generalized-vanishing: It is VC for the Laplacian and $Q=P$; the general form allows other constant-coefficient operators and test polynomials, so VC is a special case of the generalized vanishing conjecture.

Source +18 −0