Dynamic Toolbox for ETRINV

TL;DR

This paper provides a comprehensive, self-contained overview of the ETR-INV problem, including universality results and refinements, serving as a useful reference for researchers working on $orall ext{R}$-completeness and related problems.

Contribution

It offers a streamlined, detailed exposition of ETR-INV, including universality results and refinements, consolidating previous scattered information for future research.

Findings

Universality results for ETR-INV explained and proved

Refinements of earlier results provided for future research

Self-contained exposition consolidates previous scattered details

Abstract

Recently, various natural algorithmic problems have been shown to be $\exists \mathbb{R}$-complete. The reduction relied in many cases on the $\exists \mathbb{R}$-completeness of the problem ETR-INV, which served as a useful intermediate problem. Often some strengthening and modification of ETR-INV was required. This lead to a cluttered situation where no paper included all the previous details. Here, we give a streamlined exposition in a self-contained manner. We also explain and prove various universality results regarding ETR-INV. These notes should not be seen as a research paper with new results. However, we do describe some refinements of earlier results which might be useful for future research. We plan to extend and update this exposition as seems fit.