Formulating problems for real algebraic geometry

TL;DR

This paper explores how problem formulation and variable ordering significantly impact the performance of algorithms in real algebraic geometry, emphasizing the importance of tailored problem setup for practical applications.

Contribution

It surveys recent advances in problem formulation and algorithm optimization in real algebraic geometry, highlighting the importance of problem setup and algorithm choice for improved practical performance.

Findings

Variable ordering affects algorithm performance and output quality.

Problem formulation strategies can enhance algorithm efficiency.

Choosing the right algorithm variant is crucial for practical applications.

Abstract

We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both performance and output and summarise what may be done to assist with this choice. We then survey other questions of problem formulation and algorithm optimisation that have become pertinent following advances in CAD theory, including both work that is already published and work that is currently underway. With implementations now in reach of real world applications and new theory meaning algorithms are far more sensitive to the input, our thesis is that intelligently formulating problems for algorithms, and indeed choosing the correct algorithm variant for a problem, is key to improving the practical use of both quantifier elimination and symbolic real algebraic geometry in general.