Polynomial-time computation of fixed points and spectral norms for nonsingular symmetric tensors

Fix $n\in\mathord{\mathbb N}$ with $2<n$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be nonsingular. Let $\mathbf H$ denote the polynomial map whose fixed points are used to compute the spectral norm of ${\cal S}$. Nonsingular fixed-point computation conjecture. With probability $1$, all fixed points of $\mathbf H$ can be found within $\varepsilon$ approximation in time polynomial in $d$; in particular, $\|{\cal S}\|_{\sigma}$ can be computed within $\varepsilon$ approximation in time polynomial in $d$, with probability $1$. The conjecture arises from applying homotopy methods to the $(d-1)^{2n}$ fixed points associated with a nonsingular tensor. It remains open in the supplied text.