Algorithms in Real Algebraic Geometry: A Survey

TL;DR

This survey reviews the development of algorithms in real algebraic geometry, highlighting complexity, computational hardness, and recent numerical approaches for topological invariants of semi-algebraic sets.

Contribution

It provides a comprehensive overview of both classical and recent algorithms, emphasizing complexity and hardness results, and discusses numerical methods like semi-definite programming.

Findings

Effective quantifier elimination techniques are foundational.

Recent algorithms compute topological invariants of semi-algebraic sets.

Numerical approaches like semi-definite programming are gaining traction.

Abstract

We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for computing topological invariants of semi-algebraic sets. We emphasize throughout the complexity aspects of these algorithms and also discuss the computational hardness of the underlying problems. We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic sets. Even though we mostly concentrate on exact algorithms, we also discuss some numerical approaches involving semi-definite programming that have gained popularity in recent times.