Finite number of critical points on a sphere of a polynomial?

Question

Let $f$ be a polynomial in $x_1,\dots,x_n$. Suppose, I want to find all the critical points of $f$ on the sphere $\left\{ x\in \mathbb{R}^n \colon x_1^2+\dots + x_n^2=1\right\}$. Are there any obvious conditions on $f$ that ensure that the number of critical points is finite?

Answer 1

A sufficient condition is that the Hessian of the polynomial at each point of the sphere, considered as a function on the sphere as a manifold, be nonsingular. Then the polynomial is a Morse function and all critical points are isolated, hence there are finitely many since the sphere is compact.