Higher‑dimensional lower bounds for semialgebraic set size/inradius

Develop explicit, computable lower bounds on the volume or inradius of connected components of semialgebraic sets in ℝ^n (n > 1), defined by polynomial inequalities with integer coefficients, in terms of degrees and coefficient bounds—generalizing Rump’s one‑dimensional root separation bound to higher dimensions.

Background

To bound halting times for CDSs with Anosov dynamics, the authors need quantitative geometric lower bounds on encoded regions (e.g., inradius or volume) as functions of decoder complexity. In one dimension, they use a root‑separation bound from real algebraic geometry (Rump, 1979).

They highlight that no analogous bound is known in higher dimensions for semialgebraic sets defined by polynomial inequalities, which blocks obtaining explicit time‑complexity bounds for multi‑dimensional systems. Solving this would be a foundational advance connecting real algebraic geometry with dynamical complexity.