Relative perturbation theory criterion for homogeneous polynomials

Let $f(x_1,\ldots,x_n)$ be a homogeneous polynomial and let $\mathcal D$ be as in the preceding conjecture. Thus condition $(A)$ means that $f$ can be written as

$$f=\prod_m f_m,$$

where each $f_m$ is either one of $x_i$, $x_i-x_j$, or $x_i+x_j$, or satisfies that $\lvert f_m\rvert$ is bounded away from $0$ on $\mathcal D$. A polynomial problem has a relative perturbation theory when small relative changes in the inputs cause output relative errors bounded by a relative condition number, with unbounded deterioration possible only as an input relative gap tends to zero. Relative perturbation theory conjecture. Condition $(A)$ is a necessary and sufficient condition for $f$ to have a relative perturbation theory. The conjecture seeks to characterize relative stability under inexact inputs using the same factorization condition that governs accurate computation. Its resolution is not indicated in the supplied text.