## Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems

Venue: | CONTEMPORARY MATHEMATICS |

### BibTeX

@MISC{Basu_algorithmicsemi-algebraic,

author = {Saugata Basu},

title = { Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems},

year = {}

}

### OpenURL

### Abstract

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semialgebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems.

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Citation Context ...centered at the origin. Notice that these sets are semi-algebraic. For any semi-algebraic set X, we denote by X the closure of X, which is also a semi-algebraic set by the Tarksi-Seidenberg principle =-=[60, 59]-=- (see [22] for a modern treatment). The Tarksi-Seidenberg principle states that the class of semialgebraic sets is closed under linear projections or equivalently that the first order theory of the re... |

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Citation Context ...omposition were known before (see for example [49]), Collins [34] was the first to apply cylindrical algebraic decomposition in the setting of algorithmic semi-algebraic geometry. Schwartz and Sharir =-=[58]-=- realized its importance in trying to solve the motion planning problem in robotics, as well as computing topological properties of semi-algebraic sets. Variants of the basic cylindrical algebraic dec... |

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Citation Context ...he problems listed above and there have been a sequence of improvements in the complexities of such algorithms. We now have single exponential algorithms for deciding emptiness of semi-algebraic sets =-=[42, 43, 57, 16]-=-, quantifier elimination [57, 16, 6], deciding connectivity [30, 44, 31,6 SAUGATA BASU 39, 17], computing descriptions of the connected components [47, 20], computing the Euler-Poincaré characteristi... |

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Citation Context ...the Leray property is weakened to say that each t-ary intersection is (k − t + 1)-connected, then one can conclude that the nerve complex is k-connected. We refer the reader to the article by Björner =-=[26]-=- for more details. Notice that Theorem 5.34 gives a method for computing the Betti numbers of S using linear algebra from a cover of S by contractible sets for which all non-empty intersections are al... |

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Citation Context ...ition is described whose combinatorial (though not the algebraic) complexity is single exponential. This result has found several applications in discrete and computational geometry (see for instance =-=[33]-=-). Definition 4.1 (Cylindrical Algebraic Decomposition). A cylindrical algebraic decomposition of R k is a sequence S1, . . . , Sk where, for each 1 ≤ i ≤ k, Si is a finite partition of R i into semi-... |

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Citation Context ... with complexity of the order of (sd) kO(1) rather than (sd) 2k. An important motivating reason behind the search for such algorithms, is the following theorem due to Gabrielov and Vorobjov [40] (see =-=[56, 65, 53, 5]-=-, as well as the survey article [21], for work leading up to this result). Theorem 2.3. [40] For a P-semi-algebraic set S ⊂ R k , the sum of the Betti numbers of S (refer to Section 5 below for defini... |

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Citation Context ...centered at the origin. Notice that these sets are semi-algebraic. For any semi-algebraic set X, we denote by X the closure of X, which is also a semi-algebraic set by the Tarksi-Seidenberg principle =-=[60, 59]-=- (see [22] for a modern treatment). The Tarksi-Seidenberg principle states that the class of semialgebraic sets is closed under linear projections or equivalently that the first order theory of the re... |

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Citation Context ... with complexity of the order of (sd) kO(1) rather than (sd) 2k. An important motivating reason behind the search for such algorithms, is the following theorem due to Gabrielov and Vorobjov [40] (see =-=[56, 65, 53, 5]-=-, as well as the survey article [21], for work leading up to this result). Theorem 2.3. [40] For a P-semi-algebraic set S ⊂ R k , the sum of the Betti numbers of S (refer to Section 5 below for defini... |

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Citation Context ... as the study of algebraic varieties in C k or semi-algebraic sets in R k ). As a consequence of a series of astonishing theorems (conjectured by Andre Weil [68] and proved by Deligne [36, 37], Dwork =-=[38]-=- et al.), it turns out that the number of solutions of systems of polynomial equations over a finite field, Fq, in algebraic extensions of Fq, is governed by the dimensions of certain (appropriately d... |

63 |
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Citation Context ...he problems listed above and there have been a sequence of improvements in the complexities of such algorithms. We now have single exponential algorithms for deciding emptiness of semi-algebraic sets =-=[42, 43, 57, 16]-=-, quantifier elimination [57, 16, 6], deciding connectivity [30, 44, 31,6 SAUGATA BASU 39, 17], computing descriptions of the connected components [47, 20], computing the Euler-Poincaré characteristi... |

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Solving systems of polynomials inequalities in subexponential time
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Citation Context ...he problems listed above and there have been a sequence of improvements in the complexities of such algorithms. We now have single exponential algorithms for deciding emptiness of semi-algebraic sets =-=[42, 43, 57, 16]-=-, quantifier elimination [57, 16, 6], deciding connectivity [30, 44, 31,6 SAUGATA BASU 39, 17], computing descriptions of the connected components [47, 20], computing the Euler-Poincaré characteristi... |

50 | Computing roadmaps of general semi-algebraic sets
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Citation Context ...ving problems in semi-algebraic geometry are mostly based on the critical point method. This method was pioneered by several researchers including Grigoriev and Vorobjov [43, 44], Renegar [57], Canny =-=[30]-=-, Heintz, Roy and Solernò [47], Basu, Pollack and Roy [16] amongst others. In simple terms, the critical point method is nothing but a method for finding at least one point in every semi-algebraically... |

48 |
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Citation Context ... with complexity of the order of (sd) kO(1) rather than (sd) 2k. An important motivating reason behind the search for such algorithms, is the following theorem due to Gabrielov and Vorobjov [40] (see =-=[56, 65, 53, 5]-=-, as well as the survey article [21], for work leading up to this result). Theorem 2.3. [40] For a P-semi-algebraic set S ⊂ R k , the sum of the Betti numbers of S (refer to Section 5 below for defini... |

32 | New results on quantifier elimination over real closed fields and applications toconstraint databases
- Basu
(Show Context)
Citation Context ...een a sequence of improvements in the complexities of such algorithms. We now have single exponential algorithms for deciding emptiness of semi-algebraic sets [42, 43, 57, 16], quantifier elimination =-=[57, 16, 6]-=-, deciding connectivity [30, 44, 31,6 SAUGATA BASU 39, 17], computing descriptions of the connected components [47, 20], computing the Euler-Poincaré characteristic (see Section 5.3.1 below for defin... |

32 |
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Citation Context |

31 | Almost tight upper bounds for vertical decompositions in four dimensions
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Citation Context ...hose complexity is bounded by O(nk ). More efficient ways of decomposing arrangements into topological balls have been proposed. In [32] the authors provide a decomposition into O∗ (n2k−3) cells (see =-=[48]-=- for an improvement of this result in the case k = 4). However, this decomposition does not produce a cell complex and is therefore not directly useful in computing the Betti numbers of the arrangemen... |

30 |
Feasibility testing for systems of real quadratic equations
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Citation Context ...h makes it possible to avoid having double exponential complexity. Moreover, polynomial time algorithms are now known for computing some of these invariants for special classes of semi-algebraic sets =-=[3, 45, 9, 11, 24]-=-. We describe some of these new results in greater detail in Section 3. 2.4. Certain Restricted Classes of Semi-algebraic Sets. Since general semi-algebraic sets can have exponential topological compl... |

29 |
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Citation Context ...φs : ′′ Es −→ ′′ Ēs) between the associated spectral sequences (corresponding either to the row-wise or column-wise filtrations). For the precise definition of homomorphisms of spectral sequences see =-=[52]-=-. We will need the following useful fact (see [52, pp. 66]).ALGORITHMIC SEMI-ALGEBRAIC GEOMETRY AND TOPOLOGY 37 Theorem 5.43 (Comparison Theorem). If ′ φ s (respectively, ′′ φ s) is an isomorphism fo... |

28 |
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Citation Context ...any given topological space, such as a semi-algebraic set in R k , is an exceedingly difficult problem. In fact, the general problem of determining if two given spaces are homeomorphic is undecidable =-=[50]-=-. In order to get around this difficulty, mathematicians since the time of Poincaré have devised more easily computable (albeit weaker) invariants of topological spaces. One reason that cohomology gro... |

24 |
Counting connected components of a semialgebraic set in subexponential time
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- 1992
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Citation Context ...exponential complexity for solving problems in semi-algebraic geometry are mostly based on the critical point method. This method was pioneered by several researchers including Grigoriev and Vorobjov =-=[43, 44]-=-, Renegar [57], Canny [30], Heintz, Roy and Solernò [47], Basu, Pollack and Roy [16] amongst others. In simple terms, the critical point method is nothing but a method for finding at least one point i... |

22 |
Betti numbers of semialgebraic and sub-Pfaffian sets
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(Show Context)
Citation Context ...0 f (X)) C2 (W 1 f (X)) C2 (W 2 f (X)) δ δ δ �� ˜d ˜d ˜d 0 C 1 (W 0 f (X)) C1 (W 1 f (X)) C1 (W 2 f (X)) δ δ δ �� ˜d ˜d ˜d 0 C 0 (W 0 f (X)) C0 (W 1 f (X)) C0 (W 2 f (X)) δ δ δ �� 0 0 0 Theorem 5.53. =-=[41, 24]-=- For any continuous semi-algebraic surjection f : X → Y , where X and Y are open semi-algebraic subsets of R n and R m respectively (or, more generally, for any locally split continuous surjection f),... |

22 |
Description of the connected components of a semialgebraic set in single exponential time, Discrete Comput
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- 1994
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Citation Context ...ng emptiness of semi-algebraic sets [42, 43, 57, 16], quantifier elimination [57, 16, 6], deciding connectivity [30, 44, 31,6 SAUGATA BASU 39, 17], computing descriptions of the connected components =-=[47, 20]-=-, computing the Euler-Poincaré characteristic (see Section 5.3.1 below for definition) [5, 19], as well as the first few (that is, any constant number of) Betti numbers of semialgebraic sets [20, 10].... |

21 | Finding connected components b of a semialgebraic set in subexponential time - Grigoriev, Canny, et al. - 1992 |

20 |
Construction of roadmaps of semi-algebraic sets
- Gournay, Risler
- 1993
(Show Context)
Citation Context ...inside each connected component of the given set. Roadmaps were first introduced by Canny [30], but similar constructions were considered as well by Grigoriev and Vorobjov [44] and Gournay and Risler =-=[39]-=-. Our exposition below follows that in [17, 22] where the most efficient algorithm for computing roadmaps is given. The notions of pseudo-critical points and values defined above play a critical role ... |

20 |
A Vietoris mapping theorem for homotopy
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- 1957
(Show Context)
Citation Context ...x) ↦→ x. Notice that for each x ∈ A [n] , f −1 A (x) = |∆Ix|, where Ix = {i | x ∈ Ai}. In particular, f −1 −1 A (x) is contractible for each x ∈ fA (x), and it follows from the Smale-Vietoris theorem =-=[63]-=- that Lemma 5.55. The map fA is a homotopy equivalence. For ℓ ≥ 0, we will denote by hocolim≤ℓ(A) the subcomplex of hocolim(A) defined by (5.30) hocolim≤ℓ(A) = · ⋃ ∆I × AI/ ∼ I⊂[n],#I≤ℓ+2 The followin... |

19 | Computing the first few Betti numbers of semi-algebraic sets in single exponential time
- Basu
(Show Context)
Citation Context ... [47, 20], computing the Euler-Poincaré characteristic (see Section 5.3.1 below for definition) [5, 19], as well as the first few (that is, any constant number of) Betti numbers of semialgebraic sets =-=[20, 10]-=-. These algorithms answer questions about the semi-algebraic set S without obtaining a full cylindrical algebraic decomposition (see Section 4.1 below for definition), which makes it possible to avoid... |

18 | Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets
- Bürgisser, Cucker
(Show Context)
Citation Context ... In the discrete case, we might want to count the number of solutions – and this turns out to be a #P-complete problem. Recently, a #P -completeness theory has been proposed for the BSS model as well =-=[27, 28]-=- – and the natural #P complete problem in this context is computing the Euler-Poincaré characteristic of a given variety (the Euler-Poincaré characteristic being a discrete valuation is the “right” no... |

17 | Betti Numbers of semi-algebraic sets defined by quantifier-free formulae, Discrete Comput
- Gabrielov, Vorobjov
- 2005
(Show Context)
Citation Context ...algorithms with complexity of the order of (sd) kO(1) rather than (sd) 2k. An important motivating reason behind the search for such algorithms, is the following theorem due to Gabrielov and Vorobjov =-=[40]-=- (see [56, 65, 53, 5], as well as the survey article [21], for work leading up to this result). Theorem 2.3. [40] For a P-semi-algebraic set S ⊂ R k , the sum of the Betti numbers of S (refer to Secti... |

16 | Different bounds on the different Betti numbers of semi-algebraic sets
- Basu
(Show Context)
Citation Context ...ant ℓ ≥ 0, the Betti numbers bk−1(S), . . . , bk−ℓ(S), of a basic closed semi-algebraic set S ⊂ R k defined by quadratic inequalities, are polynomially bounded. The following theorem which appears in =-=[7]-=- is derived using a bound proved by Barvinok [4] on the Betti numbers of sets defined by few quadratic equations. Theorem 2.6. [7] Let R a real closed field and S ⊂ R k be defined by Then, for any ℓ ≥... |

16 |
Computing the first Betti number and the connected components of semi-algebraic sets
- Basu, Pollack, et al.
- 2005
(Show Context)
Citation Context ...ng emptiness of semi-algebraic sets [42, 43, 57, 16], quantifier elimination [57, 16, 6], deciding connectivity [30, 44, 31,6 SAUGATA BASU 39, 17], computing descriptions of the connected components =-=[47, 20]-=-, computing the Euler-Poincaré characteristic (see Section 5.3.1 below for definition) [5, 19], as well as the first few (that is, any constant number of) Betti numbers of semialgebraic sets [20, 10].... |

16 | An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation
- Oaku, Takayama
- 1999
(Show Context)
Citation Context ... one other approach towards computation of Betti numbers (of complex varieties) that we do not describe in detail in this survey. Using the theory of local cohomology and D-modules, Oaku and Takayama =-=[55]-=- and Walther [66, 67], have given explicit algorithms for computing a sub-complex of the algebraic de Rham complex of the complements of complex affine varieties (quasi-isomorphic to the full complex ... |

15 |
On Bounding the Betti Numbers and Computing the Euler
- Basu
- 1999
(Show Context)
Citation Context |

12 | Efficient algorithm for computing the Euler-Poincaré characteristic of a semialgebraic set defined by few quadratic inequalities
- Basu
(Show Context)
Citation Context ...h makes it possible to avoid having double exponential complexity. Moreover, polynomial time algorithms are now known for computing some of these invariants for special classes of semi-algebraic sets =-=[3, 45, 9, 11, 24]-=-. We describe some of these new results in greater detail in Section 3. 2.4. Certain Restricted Classes of Semi-algebraic Sets. Since general semi-algebraic sets can have exponential topological compl... |

12 | On Computing a Set of Points meeting every Semi-algebraically Connected Component of a Family of Polynomials on a Variety
- Basu, Pollack, et al.
(Show Context)
Citation Context ...ials (or infinitesimal perturbations of these polynomials). The details of this argument can be found in [22, Proposition 13.2]. The following theorem which is the best result of this kind appears in =-=[15]-=-.18 SAUGATA BASU Theorem 4.12. [15] Let Z(Q, R k ) be an algebraic set of real dimension k ′, where Q is a polynomial in R[X1, . . . , Xk] of degree at most d, and let P ⊂ R[X1, . . . , Xk] be a set ... |

10 |
Barvinok On the Betti numbers of semi-algebraic sets defined by few quadratic inequalities
- I
- 1997
(Show Context)
Citation Context ...−ℓ(S), of a basic closed semi-algebraic set S ⊂ R k defined by quadratic inequalities, are polynomially bounded. The following theorem which appears in [7] is derived using a bound proved by Barvinok =-=[4]-=- on the Betti numbers of sets defined by few quadratic equations. Theorem 2.6. [7] Let R a real closed field and S ⊂ R k be defined by Then, for any ℓ ≥ 0, P1 ≤ 0, . . . , Ps ≤ 0, deg(Pi) ≤ 2, 1 ≤ i ≤... |

10 | Computing the Betti numbers of arrangements via spectral sequences
- Basu
(Show Context)
Citation Context ... Such arrangements are ubiquitous in computational geometry (see [1]). A naive approach using triangulations would entail a complexity of O(n2k) (see Theorem 4.5 below). This problem is considered in =-=[8]-=- where an algorithm is described for computing ℓ-th n⋃ Betti number, bℓ( Si), 0 ≤ ℓ ≤ k − 1, using O(nℓ+2 ) algebraic operations. Additionally, one has to perform linear algebra on integer matrices of... |

10 |
Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time
- Basu
(Show Context)
Citation Context ... We will also denote by λi(Q), 0 ≤ i ≤ k, the eigenvalues of M, in non-decreasing order, i.e. λ0(Q) ≤ λ1(Q) ≤ · · · ≤ λk(Q). A simple argument involving diagonalizing the quadratic form Q (see [2] or =-=[11]-=-) yields that the homotopy type of the set S defined above is related to the index(Q) by Proposition 7.2. The set S is homotopy equivalent to the S k−index(Q) . Example 7.3. The following figure (Figu... |

10 | Combinatorial complexity in o-minimal geometry, Available at
- Basu
(Show Context)
Citation Context ...ti numbers of the arrangement S. A semi-algebraic set in R k is said to have constant description complexity if it can be described by a first order formula of size bounded by some constant (see also =-=[12]-=- for a more general mathematical framework). The key point which distinguishes the results in this section from those in the previous sections is that unlike before, here we are interested only in the... |

10 | Variations by complexity theorists on three themes of
- Bürgisser, Cucker
- 2004
(Show Context)
Citation Context ... In the discrete case, we might want to count the number of solutions – and this turns out to be a #P-complete problem. Recently, a #P -completeness theory has been proposed for the BSS model as well =-=[27, 28]-=- – and the natural #P complete problem in this context is computing the Euler-Poincaré characteristic of a given variety (the Euler-Poincaré characteristic being a discrete valuation is the “right” no... |

10 |
Collins: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
- E
- 1975
(Show Context)
Citation Context ...e running time cannot be bounded by a function of the size of the input which is a fixed tower of exponents. The first algorithm with a significantly better worst-case time bound was given by Collins =-=[34]-=- in 1976. His algorithm had a worst case running time doubly exponential in the number of variables. Collins’ method is to obtain a cylindrical algebraic decomposition of the given semi-algebraic set ... |

10 |
Lectures on Discrete Geometry, Springer-Verlag
- Matousek
- 2002
(Show Context)
Citation Context ...i.e. the part that depends on the degrees and number of polynomials defining each set – is bounded by a constant. This point of view, which is now standard in discrete and computational geometry (see =-=[1, 51]-=-), presents new challenges from the point of view of designing efficient algorithms for computing Betti numbers of arrangements of sets of constant description complexity. Notice that, unlike before, ... |

9 |
Pasechnik Polynomial time computing over quadratic maps I. Sampling in real algebraic sets, Computational Complexity
- Grigor’ev, V
- 2005
(Show Context)
Citation Context ...h makes it possible to avoid having double exponential complexity. Moreover, polynomial time algorithms are now known for computing some of these invariants for special classes of semi-algebraic sets =-=[3, 45, 9, 11, 24]-=-. We describe some of these new results in greater detail in Section 3. 2.4. Certain Restricted Classes of Semi-algebraic Sets. Since general semi-algebraic sets can have exponential topological compl... |

9 | Geometry of characteristic classes - Morita - 2001 |

8 | Betti number bounds, applications and algorithms - Basu, Pollack, et al. - 2005 |