Recent Advances in Real Geometric Reasoning

TL;DR

This paper reviews recent progress in real geometric reasoning, focusing on improvements to Collins' quantifier elimination method, including complex-based approaches and more targeted problem-solving techniques.

Contribution

It introduces new approaches to real quantifier elimination that enhance efficiency and focus on specific problem-solving rather than general solutions.

Findings

Development of complex-based methods for quantifier elimination

Advances towards more problem-specific solutions

Reduction in computational complexity for certain cases

Abstract

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather than "solving all problems to do with these polynomials".