The Dependence Problem for a Class of Polynomial Maps in Dimension Four (Communications in Mathematical Research, 2021)
Let $h$ be a polynomial in four variables with the singular Hessian $\mathcal{H}h$ and the gradient $∇h$ and $R$ be a nonzero relation of $∇h$. Set $H = ∇R(∇h)$. Weprove that the components of $H$ are linearly dependent when $rk\mathcal{H}h ≤ 2$ and give anecessary and sufficient condition for the components of $H$ to be linearly dependentwhen $rk\mathcal{H}h = 3$.