Efficient simplicial replacement of semi-algebraic sets

TL;DR

This paper presents an algorithm that constructs a simplicial complex approximating semi-algebraic sets up to a specified homotopy level, with complexity singly exponential in the dimension for fixed approximation levels.

Contribution

It introduces a new algorithm for simplicial approximation of semi-algebraic sets with complexity bounds that are singly exponential in the ambient dimension for fixed approximation levels.

Findings

The algorithm produces an $ ext{ell}$-equivalent simplicial complex for any semi-algebraic set.

Complexity is bounded by $(sd)^{k^{O( ext{ell})}}$, singly exponential in $k$ for fixed $ ext{ell}$.

Reduces the problem of computing low-dimensional homotopy groups to combinatorial computations on finite complexes.

Abstract

We prove that for any $\ell \geq 0$, there exists an algorithm which takes as input a description of a semi-algebraic subset $S \subset \mathbb{R}^k$ given by a quantifier-free first order formula $\phi$ in the language of the reals, and produces as output a simplicial complex $\Delta$, whose geometric realization, $|\Delta|$ is $\ell$-equivalent to $S$. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell)}}$, where $s$ is the number of polynomials appearing in the formula $\phi$, and $d$ a bound on their degrees. For fixed $\ell$, this bound is singly exponential in $k$. In particular, since $\ell$-equivalence implies that the homotopy groups up to dimension $\ell$ of $|\Delta|$ are isomorphic to those of $S$, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell$ homotopy groups of $S$ to the combinatorial problem of computing the first $\ell$ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell)}}$.