# Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

@article{Cucker2019ComputingTH, title={Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time}, author={Felipe Cucker and Peter B{\"u}rgisser and Pierre Lairez}, journal={Journal of the ACM (JACM)}, year={2019}, volume={66}, pages={1 - 30} }

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity that is doubly exponential (and this is so for almost all data).

## 28 Citations

Computing the Homology of Semialgebraic Sets. II: General Formulas

- Mathematics, Computer ScienceFound. Comput. Math.
- 2021

A numerical algorithm for computing the homology of semialgebraic sets given by Boolean formulas works in weak exponential time, which means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data.

Computing the Homology of Semialgebraic Sets. I: Lax Formulas

- Computer Science, MathematicsFound. Comput. Math.
- 2020

An algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities works in weak exponential time.

Condition and homology in semialgebraic geometry

- Mathematics, Computer Science
- 2019

This PhD thesis shows how to obtain a numerical algorithm running in single exponential time with very high probability, which improves the state-of-the-art.

Persistent homology of semi-algebraic sets

- Computer Science, Mathematics
- 2022

The algorithm is the first algorithm with singly exponential complexity for computing the barcodes up to dimension (cid:96) of a given semi-algebraic set by the sub-level sets of a Given polynomial.

Recent Advances in the Computation of the Homology of Semialgebraic Sets

- MathematicsCiE
- 2019

This article describes recent advances in the computation of the homology groups of semialgebraic sets and throws light on the main features of this technical picture, the complexity results obtained, and how the new algorithms fit into the landscape of existing results.

An Adaptive Grid Algorithm for Computing the Homology Group of Semialgebraic Set

- Computer ScienceArXiv
- 2019

This thesis will introduce the improvement of an algorithm of weak exponential time for the computation of the homology groups of an algebraic set using an adaptive grid algorithm on the unit sphere.

An Adaptive Grid Algorithm for Computing the Homology Group of Semialgebraic Set

- Computer Science
- 2019

This thesis will introduce the work on an improvement of this algorithm of weak exponential time using an adaptive grid algorithm on the unit sphere and the results will be compared to those of previous work on this topic.

Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set

- Mathematics, Computer Science
- 2021

An algorithm for computing a semi-algebraic basis for the first homology group, H1(S,F), with coefficients in a field F, of any given semi- algebraic set S ⊂ R defined by a closed formula is given, which generalizes well known algorithms having singly exponential complexity.

Computing the Volume of Compact Semi-Algebraic Sets

- Mathematics, Computer ScienceISSAC
- 2019

An algorithm which takes as input a polynomial system defining S and an integer p and returns the n-dimensional volume of S at absolute precision 2^-p and improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods.

Grid Methods in Computational Real Algebraic (and Semialgebraic) Geometry

- Mathematics, Computer Science
- 2018

A family of numerical algorithms to solve problems in real algebraic and semialgebraic geometry that is radically different from the usual complexity analysis in symbolic computation as these numerical algorithms may run forever on a thin set of ill-posed inputs.

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