[Paper Review] Two Results on Homogeneous Hessian Nilpotent Polynomials

Arno van den Essen, Wenhua Zhao|ArXiv.org|Apr 13, 2007

Advanced Differential Equations and Dynamical Systems4 references4 citations

TL;DR

This paper establishes two key results on homogeneous Hessian nilpotent (HN) polynomials: first, the vanishing conjecture holds if the projective varieties defined by the HN polynomial and the quadratic form ∑zᵢ² intersect only at regular points of the HN variety; second, the vanishing conjecture is equivalent to a stronger condition involving arbitrary polynomials f(z), where Δᵐ(fPᵐ) = 0 for large m. These results advance the Jacobian conjecture via the symmetric map reduction.

ABSTRACT

Let $z=(z_1, ..., z_n)$ and $Δ=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ 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 $Δ^m P^{m+1}(z)=0$ 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 ${\mathcal Z}_{σ_2}$ of $\mathbb C P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $σ_2(z):=\sum_{i=1}^n z_i^2$, respectively, intersect only at regular points of ${\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z- abla P$ 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 $f(z)$, $Δ^m (f(z)P(z)^m)=0$ when $m>>0$.

Motivation & Objective

• To prove the vanishing conjecture for homogeneous Hessian nilpotent (HN) polynomials under geometric conditions on their projective varieties.
• To establish an equivalence between the vanishing conjecture and a formally stronger statement involving arbitrary polynomials f(z).
• To contribute to the resolution of the Jacobian conjecture by reducing it to properties of symmetric polynomial maps F = z − ∇P.
• To clarify the role of Hessian nilpotence and Laplace operator actions in the structure of polynomial ideals and differential equations.

Proposed method

• Uses algebraic geometry to analyze the intersection of projective varieties Z_P and Z_σ₂ in ℂPⁿ⁻¹, where σ₂(z) = ∑zᵢ².
• Applies Euler's formula and the condition that common zeros correspond to fixed points of symmetric maps F = z − ∇P.
• Employs the Laplace operator Δ and its iterated action Δᵐ on Pᵐ⁺¹ to test the vanishing conjecture.
• Leverages duality in polynomial spaces and the non-degeneracy of the bilinear form B_m to show finite-dimensionality of solution spaces.
• Uses the identity Δᵐ(fPᵐ) = ∑ terms involving mixed derivatives and multinomial coefficients to analyze the stronger conjecture.
• Relies on known results from [Z] on HN polynomials and the equivalence between the Jacobian conjecture and the vanishing conjecture.

Experimental results

Research questions

• RQ1 Under what geometric conditions on the projective variety Z_P does the vanishing conjecture hold for a homogeneous HN polynomial P of degree d ≥ 4?
• RQ2 Is the vanishing conjecture equivalent to the stronger statement that Δᵐ(fPᵐ) = 0 for all polynomials f and sufficiently large m?
• RQ3 How do the fixed points of symmetric maps F = z − ∇P relate to the singularities of the variety Z_P and the vanishing of ΔᵐPᵐ⁺¹?
• RQ4 Can the vanishing conjecture be reduced to a condition on the solution space of a system of PDEs involving ∂P/∂zᵢ(D) and Δ?

Key findings

• The vanishing conjecture holds for any homogeneous HN polynomial P of degree d ≥ 4 if the projective varieties Z_P and Z_σ₂ intersect only at regular points of Z_P.
• The vanishing conjecture is equivalent to the condition that Δᵐ(f(z)P(z)ᵐ) = 0 for all polynomials f(z) and all sufficiently large m.
• If the symmetric map F = z − ∇P has no non-zero fixed point w with ∑wᵢ² = 0, then the Jacobian conjecture holds for F.
• The solution space of the system Δu = 0 and ∂P/∂zᵢ(D)u = 0 for all i is finite-dimensional if and only if the varieties Z_P and Z_σ₂ have no common non-zero zeros.
• The vanishing of ΔᵐPᵐ⁺¹ for large m follows from the finite-dimensionality of the solution space, due to degree growth in the terms.
• The stronger condition Δᵐ(fPᵐ) = 0 for all f and large m implies the vanishing conjecture, and vice versa, via explicit expansion and degree bounds on the terms.