The Nearest Zero Eigenvector of a Weakly Symmetric Tensor from a Given Point

We begin with a degree m real homogeneous polynomial in n indeterminants and bound the distance from a given n-dimensional real vector to the real vanishing of the homogeneous polynomial. We then apply these bounds to the real homogeneous polynomial associated with a nonzero m-order n-dimensional weakly symmetric tensor which has zero as an eigenvalue. We provide “nested spheres” conditions to bound the distance from a given n-dimensional real vector to the nearest zero eigenvector.