Algorithms in Real Algebraic Geometry: A Survey

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.

41 pages, 4 figures. Based on survey talk given at the Real Algebraic Geometry Conference, Rennes, June 20-24, 2011. Some references updated and some newer material added

References in corpus (3)

• Divide and Conquer Roadmap for Algebraic Sets
• Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials
• Monotone functions and maps

Cited by in corpus (10)

• Conormal Spaces and Whitney Stratifications
• Quantum algorithms for topological and geometric analysis of big data
• Low Rank Approximation with Entrywise ${\ell }_1$-Norm Error
• Topological descriptors of the parameter region of multistationarity: deciding upon connectivity
• Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions
• Decidability of Sample Complexity of PAC Learning in finite setting
• The Geometry of Memoryless Stochastic Policy Optimization in Infinite-Horizon POMDPs
• An exact algorithm for the minimum rank of a graph
• Can one design a geometry engine? On the (un)decidability of affine Euclidean geometries
• Constraints on Nonlinear Finite Dimensional Flat Systems