Lecture notes on complexity of quantifier elimination over the reals

TL;DR

This paper presents lecture notes on an algorithm with singly exponential complexity for the existential theory of the reals, emphasizing core ideas and providing an educational overview rather than detailed proofs.

Contribution

It offers an accessible exposition of a singly exponential algorithm for quantifier elimination over the reals, based on Renegar's work, tailored for MSc students.

Findings

Algorithm achieves singly exponential complexity

Provides conceptual understanding of quantifier elimination

Serves as educational resource for students

Abstract

These are lecture notes for a course I gave in mid-1990s for MSc students at the University of Bath. It presents an algorithm with singly exponential complexity for the existential theory of the reals, in the spirit of J. Renegar. The aim was to convey the main underlying ideas, so many of the proofs and finer details of algorithms are either missing or just sketched. I changed nothing in the original notes except adding references, bibliography, and correcting obvious typos.