We consider forms on the Euclidean unit sphere. We obtain obtain a simple and complete characterization of all points that satisfies the standard second-order necessary condition of optimality. It is stated solely in terms of the value of (i) f, (ii) the norm of its gradient, and (iii) the first two smallest eigenvalues of its Hessian, all evaluated at the point. In fact this property also holds for twice continuous differentiable functions that are positively homogeneous. We also characterize a class of degree-d forms with no spurious local minima on the sphere by using a property of gradient ideals in algebraic geometry.
LAAS-report October 2020 (updated)