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Efficient Computation of a Semi-Algebraic Basis of the First Homology Group of a Semi-Algebraic Set

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Abstract

Let \(\textrm{R}\) be a real closed field and \(\textrm{C}\) the algebraic closure of \(\textrm{R}\). We give an algorithm for computing a semi-algebraic basis for the first homology group, \(\textrm{H}_1(S,{\mathbb {F}})\), with coefficients in a field \({\mathbb {F}}\), of any given semi-algebraic set \(S \subset \textrm{R}^k\) defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. More precisely, if the given quantifier-free formula involves s polynomials whose degrees are bounded by d, the complexity of the algorithm is bounded by \((s d)^{k^{O(1)}}\). This algorithm generalizes well known algorithms having singly exponential complexity for computing a semi-algebraic basis of the zeroth homology group of semi-algebraic sets, which is equivalent to the problem of computing a set of points meeting every semi-algebraically connected component of the given semi-algebraic set at a unique point. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset \(\Gamma \) of the given semi-algebraic set S, such that \(\textrm{H}_q(S,\Gamma ) = 0\) for \(q=0,1\). We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety X of dimension n, there exists Zariski closed subsets

$$\begin{aligned} Z^{(n-1)} \supset \cdots \supset Z^{(1)} \supset Z^{(0)} \end{aligned}$$

with \(\dim _\textrm{C}Z^{(i)} \le i\), and \(\textrm{H}_q(X,Z^{(i)}) = 0\) for \(0 \le q \le i\). We conjecture a quantitative version of this result in the semi-algebraic category, with X and \(Z^{(i)}\) replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of \(Z^{(0)}\) and \(Z^{(1)}\) with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing \(Z^{(0)}\)).

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Notes

  1. In case \(\textrm{D}= {\mathbb {Z}}\), it is possible to deduce the bit-complexity of our algorithms in terms of the bit-sizes of the coefficients of the input polynomials, and this will agree with the classical (Turing) notion of complexity. We do not state the bit complexity separately in our algorithms, but note that it is always bounded by a polynomial in the bit-size of the input times the complexity upper bound stated in the paper. We do not count the cost of doing linear algebra over the field \({\mathbb {F}}\). However, if \({\mathbb {F}}= {\mathbb {Q}}\), then the bit complexity of doing operations over \({\mathbb {F}}\) will be bounded by the complexity upper bound stated in the paper.

  2. The reason behind insisting on “semi-algebraically” connected instead of just connected (in the Euclidean topology) is that over an arbitrary real closed field these two notions are distinct. On the other hand if \(\textrm{R}= {\mathbb {R}}\), then being semi-algebraically connected is equivalent to being connected (see for example [7, Thm. 5.22]).

  3. More modern algorithms such as those described in [8, 12] use different techniques.

  4. It is a consequence of the well-known Thom’s lemma, that the Thom encoding uniquely characterizes a root in \(\textrm{R}\) of a polynomial in \(\textrm{D}[X]\) (see for example, [7, Prop. 2.27]).

  5. Not to be confused with the homological functor \(\textrm{Ext}(\cdot ,\cdot )\).

  6. Note that without the requirement that the output formula be closed it is straight-forward to obtain a singly exponential complexity algorithm via quantifier elimination.

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Acknowledgements

We are grateful to the anonymous referees for their comments which helped to improve the paper. This research was supported by the National Science Foundation grant CCF-1910441.

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Saugata Basu

  2. Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Sarah Percival

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S. Basu was partially supported by NSF grants CCF-1618918, DMS-1620271 and CCF-1910441.

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Basu, S., Percival, S. Efficient Computation of a Semi-Algebraic Basis of the First Homology Group of a Semi-Algebraic Set. Discrete Comput Geom (2024). https://doi.org/10.1007/s00454-024-00626-0

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