# Algorithms in Real Algebraic Geometry: A Survey

arXiv: https://arxiv.org/abs/1409.1534 · HTML: https://mlsanity.com/paper/1409.1534/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2014.

## Built on by (most cited first, 5 of 5)

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- [When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time?](https://mlsanity.com/paper/2502.16912/) (2025)
- [Probabilistic Strategy Logic with Degrees of Observability](https://mlsanity.com/paper/2412.15135/) (2024)

## Abstract (arXiv)

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.