Sharp Hölder stability exponent

Determine the sharp Hölder exponent in the inverse stability estimate for separating polynomial invariants of the radial and coordinate models, including its dependence on the stabilizer strata.

Background

The paper proves that, on compact subsets, the orbit distance between truncated descriptors is bounded by a Hölder power of the discrepancy between any separating polynomial invariant vectors. At principal-stratum points where the invariant map has full-rank differential on a slice, the exponent can be taken to be one locally, whereas singular strata can force weaker behavior.

The exact optimal exponent globally and its variation across different strata are not established. The authors point to bi-Lipschitz results for related invariant constructions as a possible avenue for determining sharper bounds.