(untitled) (Communications in Mathematical Research, 2021)
Let h be a polynomial in four variables with the singular Hessian Hh and the gradient ∇h and R be a nonzero relation of ∇h. Set H = ∇R(∇h). We prove that the components of H are linearly dependent when rkHh ≤ 2 and give a necessary and sufficient condition for the components of H to be linearly dependent when rkHh = 3.