Zak Tonks: Cylindrical algebraic decomposition: algorithmic real algebraic geometry.

Abstract: Cylindrical Algebraic Decomposition is a powerful method in computational real algebraic geometry, being `merely# doubly exponential in the number of variables, rather than Tarski's original method, whose complexity was not expressible by any fixed tower of exponentials. But the original Collins method was very expensive in practice. McCallum's improvement was substantial, but had the drawback that it could occasionally fail (input not well-oriented). Recent improvements to McCallum have made the computation more problem-sensitive, rather than being a sledgehammer that solves all problems about the same polynomials. We describe these, and also the recently-verified Lazard method, which is similar to McCallum but has the advantage of never failing.