Algorithms for Computing Betti Numbers of Semi-algebraic Sets - PowerPoint PPT Presentation

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Complexity of Algorithms Double exponential vs single exponential vs polynomial (in k ) time. Problems that can be solved in single exponential time: testing emptiness, deciding connectivity, computing descriptions of the connected components, computing the Euler-Poincaré characteristic, computing the dimension of a given semi-algebraic set. Problems for which no single exponential time algorithm is known: Computing the higher Betti numbers, computing semi-algebraic triangulations, computing semi-algebraic stratifications. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Complexity of Algorithms Double exponential vs single exponential vs polynomial (in k ) time. Problems that can be solved in single exponential time: testing emptiness, deciding connectivity, computing descriptions of the connected components, computing the Euler-Poincaré characteristic, computing the dimension of a given semi-algebraic set. Problems for which no single exponential time algorithm is known: Computing the higher Betti numbers, computing semi-algebraic triangulations, computing semi-algebraic stratifications. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Complexity of Algorithms Double exponential vs single exponential vs polynomial (in k ) time. Problems that can be solved in single exponential time: testing emptiness, deciding connectivity, computing descriptions of the connected components, computing the Euler-Poincaré characteristic, computing the dimension of a given semi-algebraic set. Problems for which no single exponential time algorithm is known: Computing the higher Betti numbers, computing semi-algebraic triangulations, computing semi-algebraic stratifications. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Complexity of Algorithms Double exponential vs single exponential vs polynomial (in k ) time. Problems that can be solved in single exponential time: testing emptiness, deciding connectivity, computing descriptions of the connected components, computing the Euler-Poincaré characteristic, computing the dimension of a given semi-algebraic set. Problems for which no single exponential time algorithm is known: Computing the higher Betti numbers, computing semi-algebraic triangulations, computing semi-algebraic stratifications. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Three Main Techniques It is possible to obtain a semi-algebraic triangulation of S in doubly exponential time using cylindrical algebraic decomposition (Collins, Schwartz-Sharir). Thus, algorithms with doubly exponential complexity ( ✭ sd ✮ 2 O ✭ k ✮ ) is known for computing all the Betti numbers. Algorithms with singly exponential complexity are all based on some version of the critical point method. We do not obtain full topological information, but enough to test emptiness, to compute Euler-Poincaré characteristics, number of connected components etc. For sets defined by quadratic inequalities, there is a duality that allows us to exchange the roles of k and s , which can be exploited in certain situations. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Three Main Techniques It is possible to obtain a semi-algebraic triangulation of S in doubly exponential time using cylindrical algebraic decomposition (Collins, Schwartz-Sharir). Thus, algorithms with doubly exponential complexity ( ✭ sd ✮ 2 O ✭ k ✮ ) is known for computing all the Betti numbers. Algorithms with singly exponential complexity are all based on some version of the critical point method. We do not obtain full topological information, but enough to test emptiness, to compute Euler-Poincaré characteristics, number of connected components etc. For sets defined by quadratic inequalities, there is a duality that allows us to exchange the roles of k and s , which can be exploited in certain situations. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Three Main Techniques It is possible to obtain a semi-algebraic triangulation of S in doubly exponential time using cylindrical algebraic decomposition (Collins, Schwartz-Sharir). Thus, algorithms with doubly exponential complexity ( ✭ sd ✮ 2 O ✭ k ✮ ) is known for computing all the Betti numbers. Algorithms with singly exponential complexity are all based on some version of the critical point method. We do not obtain full topological information, but enough to test emptiness, to compute Euler-Poincaré characteristics, number of connected components etc. For sets defined by quadratic inequalities, there is a duality that allows us to exchange the roles of k and s , which can be exploited in certain situations. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Outline 1 Introduction and some recent results Statement of the Problems Complexities of Different Problems Some Recent Results 2 Outline of the Methods General Case Quadratic Case Connections to complexity theory 3 Outline of the proof Open Problems 4 Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Results about general semi-algebraic sets Computing the first Betti number. [B,Pollack,Roy, 2005] There exists an algorithm that takes as input the description of a P -semi-algebraic set S ✚ R k , and outputs b 1 ✭ S ✮ . The complexity of the algorithm is ✭ sd ✮ k O ✭ 1 ✮ . Computing the first ❵ Betti number. [B., 2005] For any fixed ❵ ❃ 0, we can compute b 0 ✭ S ✮ ❀ ✿ ✿ ✿ ❀ b ❵ ✭ S ✮ with complexity ✭ sd ✮ k O ✭ ❵ ✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Results about general semi-algebraic sets Computing the first Betti number. [B,Pollack,Roy, 2005] There exists an algorithm that takes as input the description of a P -semi-algebraic set S ✚ R k , and outputs b 1 ✭ S ✮ . The complexity of the algorithm is ✭ sd ✮ k O ✭ 1 ✮ . Computing the first ❵ Betti number. [B., 2005] For any fixed ❵ ❃ 0, we can compute b 0 ✭ S ✮ ❀ ✿ ✿ ✿ ❀ b ❵ ✭ S ✮ with complexity ✭ sd ✮ k O ✭ ❵ ✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Basic Semi-algebraic Sets Defined By Quadratic Inequalities Let S ✚ R k be a semi-algebraic set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0 ❀ with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s ✿ Such sets are in fact quite general, since every semi-algebraic set can be defined by (quantified) formulas involving only quadratic polynomials. Moreover, as in the case of general semi-algebraic sets, the Betti numbers of such sets can be exponentially large. For example, the set S ✚ R k defined by X 1 ✭ X 1 � 1 ✮ ✕ 0 ❀ ✿ ✿ ✿ ❀ X k ✭ X k � 1 ✮ ✕ 0 ❀ has b 0 ✭ S ✮ ❂ 2 k . It is NP-hard to decide whether such a set is empty. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Basic Semi-algebraic Sets Defined By Quadratic Inequalities Let S ✚ R k be a semi-algebraic set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0 ❀ with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s ✿ Such sets are in fact quite general, since every semi-algebraic set can be defined by (quantified) formulas involving only quadratic polynomials. Moreover, as in the case of general semi-algebraic sets, the Betti numbers of such sets can be exponentially large. For example, the set S ✚ R k defined by X 1 ✭ X 1 � 1 ✮ ✕ 0 ❀ ✿ ✿ ✿ ❀ X k ✭ X k � 1 ✮ ✕ 0 ❀ has b 0 ✭ S ✮ ❂ 2 k . It is NP-hard to decide whether such a set is empty. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Basic Semi-algebraic Sets Defined By Quadratic Inequalities Let S ✚ R k be a semi-algebraic set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0 ❀ with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s ✿ Such sets are in fact quite general, since every semi-algebraic set can be defined by (quantified) formulas involving only quadratic polynomials. Moreover, as in the case of general semi-algebraic sets, the Betti numbers of such sets can be exponentially large. For example, the set S ✚ R k defined by X 1 ✭ X 1 � 1 ✮ ✕ 0 ❀ ✿ ✿ ✿ ❀ X k ✭ X k � 1 ✮ ✕ 0 ❀ has b 0 ✭ S ✮ ❂ 2 k . It is NP-hard to decide whether such a set is empty. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Basic Semi-algebraic Sets Defined By Quadratic Inequalities Let S ✚ R k be a semi-algebraic set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0 ❀ with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s ✿ Such sets are in fact quite general, since every semi-algebraic set can be defined by (quantified) formulas involving only quadratic polynomials. Moreover, as in the case of general semi-algebraic sets, the Betti numbers of such sets can be exponentially large. For example, the set S ✚ R k defined by X 1 ✭ X 1 � 1 ✮ ✕ 0 ❀ ✿ ✿ ✿ ❀ X k ✭ X k � 1 ✮ ✕ 0 ❀ has b 0 ✭ S ✮ ❂ 2 k . It is NP-hard to decide whether such a set is empty. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Bounds on Betti Numbers of Sets Defined by Quadratic Inequalities Using prior results of Barvinok and inequalities derived from Mayer-Vietoris exact sequence, Theorem (B. 2003) Let S ✚ R k be defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0 ❀ with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s ✿ Then, ✥ ✦ s k O ✭ ❵ ✮ ✿ b k � ❵ ✭ S ✮ ✔ ❵ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Features of the bound For fixed ❵ ✕ 0 this gives a polynomial bound on the top ❵ Betti numbers of S (which could possibly be non-zero). Similar bounds do not hold for sets defined by polynomials of degree greater than two. For instance, the set defined by the single quartic equation, k ❳ i ✭ X i � 1 ✮ 2 � ✧ ❂ 0 ❀ X 2 i ❂ 1 will have b k � 1 ❂ 2 k , for small enough ✧ ❃ 0. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Features of the bound For fixed ❵ ✕ 0 this gives a polynomial bound on the top ❵ Betti numbers of S (which could possibly be non-zero). Similar bounds do not hold for sets defined by polynomials of degree greater than two. For instance, the set defined by the single quartic equation, k ❳ i ✭ X i � 1 ✮ 2 � ✧ ❂ 0 ❀ X 2 i ❂ 1 will have b k � 1 ❂ 2 k , for small enough ✧ ❃ 0. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Semi-algebraic sets defined by partially quadratic polynomials Let ◗ ✚ R ❬ Y 1 ❀ ✿ ✿ ✿ ❀ Y ❵ ❀ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , a family of polynomials with deg Y ✭ Q ✮ ✔ 2 ❀ deg X ✭ Q ✮ ✔ d ❀ Q ✷ ◗ ❀ ★✭ ◗ ✮ ❂ m ❀ and P ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ a family of polynomials with deg X ✭ P ✮ ✔ d ❀ P ✷ P ❀ ★✭ P ✮ ❂ s ✿ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Semi-algebraic sets defined by partially quadratic polynomials Let ◗ ✚ R ❬ Y 1 ❀ ✿ ✿ ✿ ❀ Y ❵ ❀ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , a family of polynomials with deg Y ✭ Q ✮ ✔ 2 ❀ deg X ✭ Q ✮ ✔ d ❀ Q ✷ ◗ ❀ ★✭ ◗ ✮ ❂ m ❀ and P ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ a family of polynomials with deg X ✭ P ✮ ✔ d ❀ P ✷ P ❀ ★✭ P ✮ ❂ s ✿ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Partially quadratic Theorem (B.-Pasechnik-Roy07) Let S ✚ R ❵ ✰ k be a ✭ P ❬ ◗ ✮ -closed semi-algebraic set. Then b ✭ S ✮ ✔ ❵ 2 ✭ O ✭ s ✰ ❵ ✰ m ✮ ❵ d ✮ k ✰ 2 m ✿ In particular, for m ✔ ❵ , we have b ✭ S ✮ ✔ ❵ 2 ✭ O ✭ s ✰ ❵ ✮ ❵ d ✮ k ✰ 2 m ✿ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Algorithmic Results in the Quadratic Case I Algorithm for deciding emptiness. [Barvinok,1993], [Grigoriev-Pasechnik, 2004]. There exists an algorithm which given a set of s polynomials, P ❂ ❢ P 1 ❀ ✿ ✿ ✿ ❀ P s ❣ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s , decides if S is non-empty, where S is the set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0. The complexity of the algorithm is k O ✭ s ✮ ✿ Algorithm for computing the Euler-Poincaré characteristic. [B., 2005]. There exists an algorithm for computing ✤ ✭ S ✮ whose complexity is k O ✭ s ✮ ✿ These algorithms are polynomial time for fixed s (number of polynomials). Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Algorithmic Results in the Quadratic Case I Algorithm for deciding emptiness. [Barvinok,1993], [Grigoriev-Pasechnik, 2004]. There exists an algorithm which given a set of s polynomials, P ❂ ❢ P 1 ❀ ✿ ✿ ✿ ❀ P s ❣ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s , decides if S is non-empty, where S is the set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0. The complexity of the algorithm is k O ✭ s ✮ ✿ Algorithm for computing the Euler-Poincaré characteristic. [B., 2005]. There exists an algorithm for computing ✤ ✭ S ✮ whose complexity is k O ✭ s ✮ ✿ These algorithms are polynomial time for fixed s (number of polynomials). Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems Algorithmic Results in the Quadratic Case II Polynomial time algorithm for computing top Betti numbers. [B., 2005]. We have an algorithm which given a set of s polynomials, P ❂ ❢ P 1 ❀ ✿ ✿ ✿ ❀ P s ❣ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , with deg ✭ P i ✮ ✔ 2 ❀ 1 ✔ i ✔ s , computes b k � 1 ✭ S ✮ ❀ ✿ ✿ ✿ ❀ b k � ❵ ✭ S ✮ , where S is the set defined by P 1 ✕ 0 ❀ ✿ ✿ ✿ ❀ P s ✕ 0. The complexity of the algorithm is ✥ ✦ ❵ ✰ 2 s k 2 O ✭ min ✭ ❵❀ s ✮✮ ❂ s ❵ ✰ 2 k 2 O ✭ ❵ ✮ ✿ ❳ i i ❂ 0 Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Statement of the Problems Outline of the Methods Complexities of Different Problems Connections to complexity theory Some Recent Results Open Problems More generally Partially quadratic case [B.-Pasechnik-Roy, 2007]. There exists an algorithm that takes as input the description of a ✭ P ❬ ◗ ✮ -closed semi-algebraic set S (following the previous notation ) and outputs its Betti numbers b 0 ✭ S ✮ ❀ ✿ ✿ ✿ ❀ b ❵ ✰ k � 1 ✭ S ✮ . The complexity of this algorithm is bounded by ✭✭ ❵ ✰ 1 ✮✭ s ✰ 1 ✮✭ m ✰ 1 ✮✭ d ✰ 1 ✮✮ 2 O ✭ m ✰ k ✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Outline 1 Introduction and some recent results Statement of the Problems Complexities of Different Problems Some Recent Results 2 Outline of the Methods General Case Quadratic Case Connections to complexity theory 3 Outline of the proof Open Problems 4 Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Steps in the General Case First reduce to the closed and bounded case using a recent construction of Gabrielov and Vorobjov. Instead of computing a triangulation (which we do not know how to do in single exponential time), we compute with single exponential complexity a family of closed, bounded and contractible semi-algebraic sets, ❢ X ☛ ❣ ☛ ✷ I such that, S ❂ ❬ ☛ ✷ I X ☛ . Using the Roadmap Algorithm compute the connected components of the pairwise and triple-wise intersections of the elements of the covering and their inclusion relationships. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Steps in the General Case First reduce to the closed and bounded case using a recent construction of Gabrielov and Vorobjov. Instead of computing a triangulation (which we do not know how to do in single exponential time), we compute with single exponential complexity a family of closed, bounded and contractible semi-algebraic sets, ❢ X ☛ ❣ ☛ ✷ I such that, S ❂ ❬ ☛ ✷ I X ☛ . Using the Roadmap Algorithm compute the connected components of the pairwise and triple-wise intersections of the elements of the covering and their inclusion relationships. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Steps in the General Case First reduce to the closed and bounded case using a recent construction of Gabrielov and Vorobjov. Instead of computing a triangulation (which we do not know how to do in single exponential time), we compute with single exponential complexity a family of closed, bounded and contractible semi-algebraic sets, ❢ X ☛ ❣ ☛ ✷ I such that, S ❂ ❬ ☛ ✷ I X ☛ . Using the Roadmap Algorithm compute the connected components of the pairwise and triple-wise intersections of the elements of the covering and their inclusion relationships. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Result Needed from Algebraic Topology A simple spectral sequence argument yields: Proposition Let A 1 ❀ ✿ ✿ ✿ ❀ A n be sub-complexes of a finite simplicial complex A such that A ❂ A 1 ❬ ✁ ✁ ✁ ❬ A n and each A i is acyclic. Then, b 1 ✭ A ✮ ❂ dim ✭ Ker ✭ ✍ 2 ✮✮ � dim ✭ Im ✭ ✍ 1 ✮✮ , with ❨ ❨ ❨ ✍ 1 ✍ 2 H 0 ✭ A i ✮ H 0 ✭ A i ❀ j ✮ H 0 ✭ A i ❀ j ❀❵ ✮ � ✦ � ✦ i i ❁ j i ❁ j ❁❵ The homomorphisms ✍ i are induced by generalized restrictions. Corresponding result needed for computing the higher Betti numbers is much more complicated. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Connecting paths Given a semi-algebraic set S ✚ R k , x ❀ y ✷ S , there exists an algorithm (Roadmap) with single exponential complexity which can decide whether x and y are in the same connected component of S and if so output a semi-algebraic path connecting x to y in S . Fix a finite set of distinguished points in every connected component of S and for x ✷ S , let ✌ ✭ x ✮ denote the connecting path computed by the algorithm connecting x to a distinguished point. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Connecting paths Given a semi-algebraic set S ✚ R k , x ❀ y ✷ S , there exists an algorithm (Roadmap) with single exponential complexity which can decide whether x and y are in the same connected component of S and if so output a semi-algebraic path connecting x to y in S . Fix a finite set of distinguished points in every connected component of S and for x ✷ S , let ✌ ✭ x ✮ denote the connecting path computed by the algorithm connecting x to a distinguished point. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Important Property of Connecting Path The connecting path ✌ ✭ x ✮ consists of two consecutive parts, ✌ 0 ✭ x ✮ and � 1 ✭ x ✮ . The path ✌ 0 ✭ x ✮ is contained in RM ✭ S ✮ and the path � 1 ✭ x ✮ is contained in S x 1 . Moreover, � 1 ✭ x ✮ can again be decomposed into two parts, ✌ 1 ✭ x ✮ and � 2 ✭ x ✮ with � 2 ✭ x ✮ contained in S x 1 ❀ x 2 and so on. If y ❂ ✭ y 1 ❀ ✿ ✿ ✿ ❀ y k ✮ ✷ S is another point such that x 1 ✻ ❂ y 1 , then the images of � 1 ✭ x ✮ and � 1 ✭ y ✮ are disjoint. If the image of ✌ 0 ✭ y ✮ (which is contained in S ) follows the same sequence of curve segments as ✌ 0 ✭ x ✮ starting at p , then the images of the paths ✌ ✭ x ✮ and ✌ ✭ y ✮ has the property that they are identical upto a point and they are disjoint after it. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Important Property of Connecting Path The connecting path ✌ ✭ x ✮ consists of two consecutive parts, ✌ 0 ✭ x ✮ and � 1 ✭ x ✮ . The path ✌ 0 ✭ x ✮ is contained in RM ✭ S ✮ and the path � 1 ✭ x ✮ is contained in S x 1 . Moreover, � 1 ✭ x ✮ can again be decomposed into two parts, ✌ 1 ✭ x ✮ and � 2 ✭ x ✮ with � 2 ✭ x ✮ contained in S x 1 ❀ x 2 and so on. If y ❂ ✭ y 1 ❀ ✿ ✿ ✿ ❀ y k ✮ ✷ S is another point such that x 1 ✻ ❂ y 1 , then the images of � 1 ✭ x ✮ and � 1 ✭ y ✮ are disjoint. If the image of ✌ 0 ✭ y ✮ (which is contained in S ) follows the same sequence of curve segments as ✌ 0 ✭ x ✮ starting at p , then the images of the paths ✌ ✭ x ✮ and ✌ ✭ y ✮ has the property that they are identical upto a point and they are disjoint after it. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Important Property of Connecting Path The connecting path ✌ ✭ x ✮ consists of two consecutive parts, ✌ 0 ✭ x ✮ and � 1 ✭ x ✮ . The path ✌ 0 ✭ x ✮ is contained in RM ✭ S ✮ and the path � 1 ✭ x ✮ is contained in S x 1 . Moreover, � 1 ✭ x ✮ can again be decomposed into two parts, ✌ 1 ✭ x ✮ and � 2 ✭ x ✮ with � 2 ✭ x ✮ contained in S x 1 ❀ x 2 and so on. If y ❂ ✭ y 1 ❀ ✿ ✿ ✿ ❀ y k ✮ ✷ S is another point such that x 1 ✻ ❂ y 1 , then the images of � 1 ✭ x ✮ and � 1 ✭ y ✮ are disjoint. If the image of ✌ 0 ✭ y ✮ (which is contained in S ) follows the same sequence of curve segments as ✌ 0 ✭ x ✮ starting at p , then the images of the paths ✌ ✭ x ✮ and ✌ ✭ y ✮ has the property that they are identical upto a point and they are disjoint after it. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Schematic Picture γ 0 ,m − 1 y X 1 γ 0 ,m − 2 Γ 1 ( y ) x Γ 1 ( x ) γ 0 ,m γ 0 ,m ( b 0 ,m ( y )) γ 0 ,m ( b 0 ,m ( x )) Figure: The connecting path �✭ x ✮ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Parametrized Paths: Precise Definition A parametrized path ✌ is a continuous semi-algebraic mapping, V ✚ R k ✰ 1 ✦ R k , a semi-algebraic continuous function ❵ ✿ U ✦ ❬ 0 ❀ ✰ ✶ ✮ , with U ❂ ✙ 1 ✿✿✿ k ✭ V ✮ ✚ R k , and a in R k , such that V ❂ ❢ ✭ x ❀ t ✮ ❥ x ✷ U ❀ 0 ✔ t ✔ ❵ ✭ x ✮ ❣ , 1 ✽ x ✷ U ❀ ✌ ✭ x ❀ 0 ✮ ❂ a , 2 ✽ x ✷ U ❀ ✌ ✭ x ❀ ❵ ✭ x ✮✮ ❂ x , 3 ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ ❵ ✭ x ✮❪ ❀ ✽ t ✷ ❬ 0 ❀ ❵ ✭ y ✮❪ 4 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ t ✮ ✮ s ❂ t ✮ , ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ min ✭ ❵ ✭ x ✮ ❀ ❵ ✭ y ✮✮❪ 5 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ s ✮ ✮ ✽ t ✔ s ❀ ✌ ✭ x ❀ t ✮ ❂ ✌ ✭ y ❀ t ✮✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Parametrized Paths: Precise Definition A parametrized path ✌ is a continuous semi-algebraic mapping, V ✚ R k ✰ 1 ✦ R k , a semi-algebraic continuous function ❵ ✿ U ✦ ❬ 0 ❀ ✰ ✶ ✮ , with U ❂ ✙ 1 ✿✿✿ k ✭ V ✮ ✚ R k , and a in R k , such that V ❂ ❢ ✭ x ❀ t ✮ ❥ x ✷ U ❀ 0 ✔ t ✔ ❵ ✭ x ✮ ❣ , 1 ✽ x ✷ U ❀ ✌ ✭ x ❀ 0 ✮ ❂ a , 2 ✽ x ✷ U ❀ ✌ ✭ x ❀ ❵ ✭ x ✮✮ ❂ x , 3 ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ ❵ ✭ x ✮❪ ❀ ✽ t ✷ ❬ 0 ❀ ❵ ✭ y ✮❪ 4 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ t ✮ ✮ s ❂ t ✮ , ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ min ✭ ❵ ✭ x ✮ ❀ ❵ ✭ y ✮✮❪ 5 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ s ✮ ✮ ✽ t ✔ s ❀ ✌ ✭ x ❀ t ✮ ❂ ✌ ✭ y ❀ t ✮✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Parametrized Paths: Precise Definition A parametrized path ✌ is a continuous semi-algebraic mapping, V ✚ R k ✰ 1 ✦ R k , a semi-algebraic continuous function ❵ ✿ U ✦ ❬ 0 ❀ ✰ ✶ ✮ , with U ❂ ✙ 1 ✿✿✿ k ✭ V ✮ ✚ R k , and a in R k , such that V ❂ ❢ ✭ x ❀ t ✮ ❥ x ✷ U ❀ 0 ✔ t ✔ ❵ ✭ x ✮ ❣ , 1 ✽ x ✷ U ❀ ✌ ✭ x ❀ 0 ✮ ❂ a , 2 ✽ x ✷ U ❀ ✌ ✭ x ❀ ❵ ✭ x ✮✮ ❂ x , 3 ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ ❵ ✭ x ✮❪ ❀ ✽ t ✷ ❬ 0 ❀ ❵ ✭ y ✮❪ 4 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ t ✮ ✮ s ❂ t ✮ , ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ min ✭ ❵ ✭ x ✮ ❀ ❵ ✭ y ✮✮❪ 5 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ s ✮ ✮ ✽ t ✔ s ❀ ✌ ✭ x ❀ t ✮ ❂ ✌ ✭ y ❀ t ✮✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Parametrized Paths: Precise Definition A parametrized path ✌ is a continuous semi-algebraic mapping, V ✚ R k ✰ 1 ✦ R k , a semi-algebraic continuous function ❵ ✿ U ✦ ❬ 0 ❀ ✰ ✶ ✮ , with U ❂ ✙ 1 ✿✿✿ k ✭ V ✮ ✚ R k , and a in R k , such that V ❂ ❢ ✭ x ❀ t ✮ ❥ x ✷ U ❀ 0 ✔ t ✔ ❵ ✭ x ✮ ❣ , 1 ✽ x ✷ U ❀ ✌ ✭ x ❀ 0 ✮ ❂ a , 2 ✽ x ✷ U ❀ ✌ ✭ x ❀ ❵ ✭ x ✮✮ ❂ x , 3 ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ ❵ ✭ x ✮❪ ❀ ✽ t ✷ ❬ 0 ❀ ❵ ✭ y ✮❪ 4 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ t ✮ ✮ s ❂ t ✮ , ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ min ✭ ❵ ✭ x ✮ ❀ ❵ ✭ y ✮✮❪ 5 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ s ✮ ✮ ✽ t ✔ s ❀ ✌ ✭ x ❀ t ✮ ❂ ✌ ✭ y ❀ t ✮✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Parametrized Paths: Precise Definition A parametrized path ✌ is a continuous semi-algebraic mapping, V ✚ R k ✰ 1 ✦ R k , a semi-algebraic continuous function ❵ ✿ U ✦ ❬ 0 ❀ ✰ ✶ ✮ , with U ❂ ✙ 1 ✿✿✿ k ✭ V ✮ ✚ R k , and a in R k , such that V ❂ ❢ ✭ x ❀ t ✮ ❥ x ✷ U ❀ 0 ✔ t ✔ ❵ ✭ x ✮ ❣ , 1 ✽ x ✷ U ❀ ✌ ✭ x ❀ 0 ✮ ❂ a , 2 ✽ x ✷ U ❀ ✌ ✭ x ❀ ❵ ✭ x ✮✮ ❂ x , 3 ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ ❵ ✭ x ✮❪ ❀ ✽ t ✷ ❬ 0 ❀ ❵ ✭ y ✮❪ 4 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ t ✮ ✮ s ❂ t ✮ , ✽ x ✷ U ❀ ✽ y ✷ U ❀ ✽ s ✷ ❬ 0 ❀ min ✭ ❵ ✭ x ✮ ❀ ❵ ✭ y ✮✮❪ 5 ✭ ✌ ✭ x ❀ s ✮ ❂ ✌ ✭ y ❀ s ✮ ✮ ✽ t ✔ s ❀ ✌ ✭ x ❀ t ✮ ❂ ✌ ✭ y ❀ t ✮✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Schematic Picture a U Figure: A Parametrized Path Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Useful property of Parametrized Paths Proposition Let ✌ ✿ V ✦ R k be a parametrized path such that U ❂ ✙ 1 ✿✿✿ k ✭ V ✮ is closed and bounded. Then, the image of ✌ is semi-algebraically contractible. The images of the parametrized paths are the building blocks in the construction of the covering by contractible sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Computing Parametrized Paths Given a closed and bounded semi-algebraic set S ✚ R k , there exists an algorithm which outputs, a finite set of t polynomials ❆ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , for every ✛ ✷ Sign ✭ ❆ ❀ S ✮ , a parametrized path ✌ ✛ ✿ V ✛ ✦ R k , with base U ✛ ❂ ❘ ✭ ✛ ✮ , such that for each y ✷ ❘ ✭ ✛ ✮ , Im ✌ ✛ ✭ y ❀ ✁ ✮ is a semi-algebraic path which connects the point y to a distinguished point a ✛ . The complexity is single exponential. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Computing Parametrized Paths Given a closed and bounded semi-algebraic set S ✚ R k , there exists an algorithm which outputs, a finite set of t polynomials ❆ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , for every ✛ ✷ Sign ✭ ❆ ❀ S ✮ , a parametrized path ✌ ✛ ✿ V ✛ ✦ R k , with base U ✛ ❂ ❘ ✭ ✛ ✮ , such that for each y ✷ ❘ ✭ ✛ ✮ , Im ✌ ✛ ✭ y ❀ ✁ ✮ is a semi-algebraic path which connects the point y to a distinguished point a ✛ . The complexity is single exponential. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Computing Parametrized Paths Given a closed and bounded semi-algebraic set S ✚ R k , there exists an algorithm which outputs, a finite set of t polynomials ❆ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , for every ✛ ✷ Sign ✭ ❆ ❀ S ✮ , a parametrized path ✌ ✛ ✿ V ✛ ✦ R k , with base U ✛ ❂ ❘ ✭ ✛ ✮ , such that for each y ✷ ❘ ✭ ✛ ✮ , Im ✌ ✛ ✭ y ❀ ✁ ✮ is a semi-algebraic path which connects the point y to a distinguished point a ✛ . The complexity is single exponential. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Computing Parametrized Paths Given a closed and bounded semi-algebraic set S ✚ R k , there exists an algorithm which outputs, a finite set of t polynomials ❆ ✚ R ❬ X 1 ❀ ✿ ✿ ✿ ❀ X k ❪ , for every ✛ ✷ Sign ✭ ❆ ❀ S ✮ , a parametrized path ✌ ✛ ✿ V ✛ ✦ R k , with base U ✛ ❂ ❘ ✭ ✛ ✮ , such that for each y ✷ ❘ ✭ ✛ ✮ , Im ✌ ✛ ✭ y ❀ ✁ ✮ is a semi-algebraic path which connects the point y to a distinguished point a ✛ . The complexity is single exponential. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Ensuring Contractibility For each ✛ ✷ Sign ✭ ❆ ❀ S ✮ since ❘ ✭ ✛❀ S ✮ is not necessarily closed and bounded, Im ✌ ✛ might not be contractible. In order to ensure contractibility, we restrict the base of ✌ ✛ to a slightly smaller set which is closed, using infinitesimals. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Ensuring the Covering Property The images of the parametrized paths obtained after shrinking their bases do not necessarily cover S . We enlarge them, preserving contractibility, to recover a covering of S . It is necessary to use 2 t infinitesimals in the shrinking and enlargement process to work correctly. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Ensuring the Covering Property The images of the parametrized paths obtained after shrinking their bases do not necessarily cover S . We enlarge them, preserving contractibility, to recover a covering of S . It is necessary to use 2 t infinitesimals in the shrinking and enlargement process to work correctly. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Ensuring the Covering Property The images of the parametrized paths obtained after shrinking their bases do not necessarily cover S . We enlarge them, preserving contractibility, to recover a covering of S . It is necessary to use 2 t infinitesimals in the shrinking and enlargement process to work correctly. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Schematic Picture Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Schematic Picture Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Schematic Picture Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Outline 1 Introduction and some recent results Statement of the Problems Complexities of Different Problems Some Recent Results 2 Outline of the Methods General Case Quadratic Case Connections to complexity theory 3 Outline of the proof Open Problems 4 Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Ideas Consider S as the intersection of the individual sets, S i defined by P i ✕ 0. The top dimensional homology groups of S are isomorphic to those of the total complex associated to a suitable truncation of the Mayer-Vietoris double complex. The terms appearing in the truncated complex depend on the unions of the S i ’s taken at most ❵ ✰ 2 at a time. There ✁ ❂ O ✭ s ❵ ✰ 2 ✮ such sets. � s are at most P ❵ ✰ 2 j ❂ 1 j Moreover, for such semi-algebraic sets we are able to compute in polynomial (in k ) time a cell complex, whose homology groups are isomorphic to those of these sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Ideas Consider S as the intersection of the individual sets, S i defined by P i ✕ 0. The top dimensional homology groups of S are isomorphic to those of the total complex associated to a suitable truncation of the Mayer-Vietoris double complex. The terms appearing in the truncated complex depend on the unions of the S i ’s taken at most ❵ ✰ 2 at a time. There ✁ ❂ O ✭ s ❵ ✰ 2 ✮ such sets. � s are at most P ❵ ✰ 2 j ❂ 1 j Moreover, for such semi-algebraic sets we are able to compute in polynomial (in k ) time a cell complex, whose homology groups are isomorphic to those of these sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Ideas Consider S as the intersection of the individual sets, S i defined by P i ✕ 0. The top dimensional homology groups of S are isomorphic to those of the total complex associated to a suitable truncation of the Mayer-Vietoris double complex. The terms appearing in the truncated complex depend on the unions of the S i ’s taken at most ❵ ✰ 2 at a time. There ✁ ❂ O ✭ s ❵ ✰ 2 ✮ such sets. � s are at most P ❵ ✰ 2 j ❂ 1 j Moreover, for such semi-algebraic sets we are able to compute in polynomial (in k ) time a cell complex, whose homology groups are isomorphic to those of these sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Main Ideas Consider S as the intersection of the individual sets, S i defined by P i ✕ 0. The top dimensional homology groups of S are isomorphic to those of the total complex associated to a suitable truncation of the Mayer-Vietoris double complex. The terms appearing in the truncated complex depend on the unions of the S i ’s taken at most ❵ ✰ 2 at a time. There ✁ ❂ O ✭ s ❵ ✰ 2 ✮ such sets. � s are at most P ❵ ✰ 2 j ❂ 1 j Moreover, for such semi-algebraic sets we are able to compute in polynomial (in k ) time a cell complex, whose homology groups are isomorphic to those of these sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

� Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Topology of Unions For quadratic forms P 1 ❀ ✿ ✿ ✿ ❀ P s , we denote by P ❂ ✭ P 1 ❀ ✿ ✿ ✿ ❀ P s ✮ ✿ R k ✰ 1 ✦ R s , the map defined by the polynomials P 1 ❀ ✿ ✿ ✿ ❀ P s . Let A ❂ ❬ P ✷P ❢ x ✷ S k ❥ P ✭ x ✮ ✔ 0 ❣ ❀ and ✡ ❂ ❢ ✦ ✷ S s ❥ ✦ i ✔ 0 ❀ 1 ✔ i ✔ s ❣ ✿ For ✦ ✷ ✡ let ✦ P ❂ P s i ❂ 1 ✦ i P i ❀ and let B ❂ ❢ ✭ ✦❀ x ✮ ❥ ✦ ✷ ✡ ❀ x ✷ S k and ✦ P ✭ x ✮ ✕ 0 ❣ ✿ B � � � � � ✣ 2 � � � � � � ✣ 1 � � � � � S k ✡ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

� Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Topology of Unions For quadratic forms P 1 ❀ ✿ ✿ ✿ ❀ P s , we denote by P ❂ ✭ P 1 ❀ ✿ ✿ ✿ ❀ P s ✮ ✿ R k ✰ 1 ✦ R s , the map defined by the polynomials P 1 ❀ ✿ ✿ ✿ ❀ P s . Let A ❂ ❬ P ✷P ❢ x ✷ S k ❥ P ✭ x ✮ ✔ 0 ❣ ❀ and ✡ ❂ ❢ ✦ ✷ S s ❥ ✦ i ✔ 0 ❀ 1 ✔ i ✔ s ❣ ✿ For ✦ ✷ ✡ let ✦ P ❂ P s i ❂ 1 ✦ i P i ❀ and let B ❂ ❢ ✭ ✦❀ x ✮ ❥ ✦ ✷ ✡ ❀ x ✷ S k and ✦ P ✭ x ✮ ✕ 0 ❣ ✿ B � � � � � ✣ 2 � � � � � � ✣ 1 � � � � � S k ✡ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

� Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Topology of Unions For quadratic forms P 1 ❀ ✿ ✿ ✿ ❀ P s , we denote by P ❂ ✭ P 1 ❀ ✿ ✿ ✿ ❀ P s ✮ ✿ R k ✰ 1 ✦ R s , the map defined by the polynomials P 1 ❀ ✿ ✿ ✿ ❀ P s . Let A ❂ ❬ P ✷P ❢ x ✷ S k ❥ P ✭ x ✮ ✔ 0 ❣ ❀ and ✡ ❂ ❢ ✦ ✷ S s ❥ ✦ i ✔ 0 ❀ 1 ✔ i ✔ s ❣ ✿ For ✦ ✷ ✡ let ✦ P ❂ P s i ❂ 1 ✦ i P i ❀ and let B ❂ ❢ ✭ ✦❀ x ✮ ❥ ✦ ✷ ✡ ❀ x ✷ S k and ✦ P ✭ x ✮ ✕ 0 ❣ ✿ B � � � � � ✣ 2 � � � � � � ✣ 1 � � � � � S k ✡ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

� Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Topology of Unions For quadratic forms P 1 ❀ ✿ ✿ ✿ ❀ P s , we denote by P ❂ ✭ P 1 ❀ ✿ ✿ ✿ ❀ P s ✮ ✿ R k ✰ 1 ✦ R s , the map defined by the polynomials P 1 ❀ ✿ ✿ ✿ ❀ P s . Let A ❂ ❬ P ✷P ❢ x ✷ S k ❥ P ✭ x ✮ ✔ 0 ❣ ❀ and ✡ ❂ ❢ ✦ ✷ S s ❥ ✦ i ✔ 0 ❀ 1 ✔ i ✔ s ❣ ✿ For ✦ ✷ ✡ let ✦ P ❂ P s i ❂ 1 ✦ i P i ❀ and let B ❂ ❢ ✭ ✦❀ x ✮ ❥ ✦ ✷ ✡ ❀ x ✷ S k and ✦ P ✭ x ✮ ✕ 0 ❣ ✿ B � � � � � ✣ 2 � � � � � � ✣ 1 � � � � � S k ✡ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Property of ✣ 2 Proposition (Agrachev) The map ✣ 2 gives a homotopy equivalence between B and ✣ 2 ✭ B ✮ ❂ A. Thus, in order to compute a complex quasi-isomorphic to C ✎ ✭ A ✮ it suffices to construct one quasi-isomorphic to C ✎ ✭ B ✮ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods General Case Connections to complexity theory Quadratic Case Open Problems Property of ✣ 1 Proposition For ✦ ✷ ✡ , ✣ � 1 1 ✭ ✦ ✮ is homotopy equivalent to the sphere S k � index ✭ ✦ P ✮ , where index ✭ ✦ P ✮ is the number of negative eigenvalues of the quadratic form ✦ P. Using this Proposition and an index invariant triangulation of ✡ , it is possible to construct a complex quasi-isomorphic to C ✎ ✭ B ✮ . Complexity is doubly exponential in ❵ . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Two classes of problems in algorithmic semi-algebraic geometry The most important algorithmic problems studied in this area fall into two broad sub-classes: the problem of quantifier elimination, and its special cases 1 such as deciding a sentence in the first order theory of reals, or deciding emptiness of semi-algebraic sets. the problem of computing topological invariants of 2 semi-algebraic sets, such as the number of connected components, Euler-Poincaré characteristic, and more generally all the Betti numbers of semi-algebraic sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Two classes of problems in algorithmic semi-algebraic geometry The most important algorithmic problems studied in this area fall into two broad sub-classes: the problem of quantifier elimination, and its special cases 1 such as deciding a sentence in the first order theory of reals, or deciding emptiness of semi-algebraic sets. the problem of computing topological invariants of 2 semi-algebraic sets, such as the number of connected components, Euler-Poincaré characteristic, and more generally all the Betti numbers of semi-algebraic sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Two classes of problems in algorithmic semi-algebraic geometry The most important algorithmic problems studied in this area fall into two broad sub-classes: the problem of quantifier elimination, and its special cases 1 such as deciding a sentence in the first order theory of reals, or deciding emptiness of semi-algebraic sets. the problem of computing topological invariants of 2 semi-algebraic sets, such as the number of connected components, Euler-Poincaré characteristic, and more generally all the Betti numbers of semi-algebraic sets. Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Compact general decision problem with at most ✦ quantifier alternations ( GDP c ✦ ) Input. A sentence ✟ in the first order theory of R ✭ Q 1 X 1 ✷ S k 1 ✮ ✁ ✁ ✁ ✭ Q ✦ X ✦ ✷ S k ✦ ✮ ✣ ✭ X 1 ❀ ✿ ✿ ✿ ❀ X ✦ ✮ ❀ where for each i ❀ 1 ✔ i ✔ ✦ , X i ❂ ✭ X i 0 ❀ ✿ ✿ ✿ ❀ X i k i ✮ is a block of k i ✰ 1 variables, Q i ✷ ❢✾ ❀ ✽❣ , with Q j ✻ ❂ Q j ✰ 1 ❀ 1 ✔ j ❁ ✦ , and ✣ is a quantifier-free formula defining a closed semi-algebraic subset S of S k 1 ✂ ✁ ✁ ✁ ✂ S k ✦ . Output. True or False depending on whether ✟ is true or false in the first order theory of R . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Compact general decision problem with at most ✦ quantifier alternations ( GDP c ✦ ) Input. A sentence ✟ in the first order theory of R ✭ Q 1 X 1 ✷ S k 1 ✮ ✁ ✁ ✁ ✭ Q ✦ X ✦ ✷ S k ✦ ✮ ✣ ✭ X 1 ❀ ✿ ✿ ✿ ❀ X ✦ ✮ ❀ where for each i ❀ 1 ✔ i ✔ ✦ , X i ❂ ✭ X i 0 ❀ ✿ ✿ ✿ ❀ X i k i ✮ is a block of k i ✰ 1 variables, Q i ✷ ❢✾ ❀ ✽❣ , with Q j ✻ ❂ Q j ✰ 1 ❀ 1 ✔ j ❁ ✦ , and ✣ is a quantifier-free formula defining a closed semi-algebraic subset S of S k 1 ✂ ✁ ✁ ✁ ✂ S k ✦ . Output. True or False depending on whether ✟ is true or false in the first order theory of R . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Computing the Poincaré polynomial of semi-algebraic sets ( Poincaré ) Input. A quantifier-free formula defining a semi-algebraic set S ✚ R k . ❂ P def i b i ✭ S ✮ T i . Output. The Poincaré polynomial P S ✭ T ✮ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Computing the Poincaré polynomial of semi-algebraic sets ( Poincaré ) Input. A quantifier-free formula defining a semi-algebraic set S ✚ R k . ❂ P def i b i ✭ S ✮ T i . Output. The Poincaré polynomial P S ✭ T ✮ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Reduction Result Theorem (B-Zell09) For every ✦ ❃ 0 , there is a polynomial time reduction of GDP c ✦ to Poincaré . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Analogy with Toda’s Theorem in discrete complexity theory The previous result allows us to deduce in an appropriate model (Blum-Shub-Smale) an analog of Toda’s theorem, which is a seminal result in discrete complexity theory and gives the following inclusion. PH ✚ P ★ P Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Analogy with Toda’s Theorem in discrete complexity theory The previous result allows us to deduce in an appropriate model (Blum-Shub-Smale) an analog of Toda’s theorem, which is a seminal result in discrete complexity theory and gives the following inclusion. PH ✚ P ★ P Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Analogy with Toda’s Theorem in discrete complexity theory The previous result allows us to deduce in an appropriate model (Blum-Shub-Smale) an analog of Toda’s theorem, which is a seminal result in discrete complexity theory and gives the following inclusion. PH ✚ P ★ P Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Analogy with Toda’s Theorem in discrete complexity theory (cont.) The classes PH and ★ P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class ★ P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Analogy with Toda’s Theorem in discrete complexity theory (cont.) The classes PH and ★ P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class ★ P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Analogy with Toda’s Theorem in discrete complexity theory (cont.) The classes PH and ★ P appearing in the two sides of the inclusion in Toda’s Theorem can be identified with the two broad classes of problems in algorithmic semi-algebraic geometry; the class PH with the problem of deciding sentences with a fixed number of quantifier alternations; the class ★ P with the problem of computing topological invariants of semi-algebraic sets, namely their Betti numbers, which generalizes the notion of cardinality for finite sets; Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Outline 1 Introduction and some recent results Statement of the Problems Complexities of Different Problems Some Recent Results 2 Outline of the Methods General Case Quadratic Case Connections to complexity theory 3 Outline of the proof Open Problems 4 Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Summary of the Main Idea Our main tool is a topological construction which given a semi-algebraic set S ✚ R m ✰ n , p ✕ 0, and ✙ Y ✿ R m ✰ n ✦ R n denoting the projection along (say) the Y -co-ordinates, constructs efficiently a semi-algebraic set, D p Y ✭ S ✮ , such that b i ✭ ✙ Y ✭ S ✮✮ ❂ b i ✭ D p Y ✭ S ✮✮ ❀ 0 ✔ i ❁ p ✿ Notice that even if there exists an efficient (i.e. polynomial time) algorithm for checking membership in S , the same need not be true for the image ✙ Y ✭ S ✮ . A second topological ingredient is Alexander-Lefshetz duality which relates the Betti numbers of a compact subset K of the sphere S n with those of S n � K . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Summary of the Main Idea Our main tool is a topological construction which given a semi-algebraic set S ✚ R m ✰ n , p ✕ 0, and ✙ Y ✿ R m ✰ n ✦ R n denoting the projection along (say) the Y -co-ordinates, constructs efficiently a semi-algebraic set, D p Y ✭ S ✮ , such that b i ✭ ✙ Y ✭ S ✮✮ ❂ b i ✭ D p Y ✭ S ✮✮ ❀ 0 ✔ i ❁ p ✿ Notice that even if there exists an efficient (i.e. polynomial time) algorithm for checking membership in S , the same need not be true for the image ✙ Y ✭ S ✮ . A second topological ingredient is Alexander-Lefshetz duality which relates the Betti numbers of a compact subset K of the sphere S n with those of S n � K . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Summary of the Main Idea Our main tool is a topological construction which given a semi-algebraic set S ✚ R m ✰ n , p ✕ 0, and ✙ Y ✿ R m ✰ n ✦ R n denoting the projection along (say) the Y -co-ordinates, constructs efficiently a semi-algebraic set, D p Y ✭ S ✮ , such that b i ✭ ✙ Y ✭ S ✮✮ ❂ b i ✭ D p Y ✭ S ✮✮ ❀ 0 ✔ i ❁ p ✿ Notice that even if there exists an efficient (i.e. polynomial time) algorithm for checking membership in S , the same need not be true for the image ✙ Y ✭ S ✮ . A second topological ingredient is Alexander-Lefshetz duality which relates the Betti numbers of a compact subset K of the sphere S n with those of S n � K . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Topological Join The join J ✭ X ❀ Y ✮ of two topological spaces X and Y is defined by ❂ X ✂ Y ✂ ✁ 1 ❂ ✘ ❀ def J ✭ X ❀ Y ✮ where ✭ x ❀ y ❀ t 0 ❀ t 1 ✮ ✘ ✭ x ✵ ❀ y ✵ ❀ t 0 ❀ t 1 ✮ if t 0 ❂ 1 ❀ x ❂ x ✵ or t 1 ❂ 1 ❀ y ❂ y ✵ . Intuitively, J ✭ X ❀ Y ✮ is obtained by joining each point of X with each point of Y by a unit interval. Example: J ✭ S m ❀ S n ✮ ✘ ❂ S m ✰ n ✰ 1 ✿ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Topological Join The join J ✭ X ❀ Y ✮ of two topological spaces X and Y is defined by ❂ X ✂ Y ✂ ✁ 1 ❂ ✘ ❀ def J ✭ X ❀ Y ✮ where ✭ x ❀ y ❀ t 0 ❀ t 1 ✮ ✘ ✭ x ✵ ❀ y ✵ ❀ t 0 ❀ t 1 ✮ if t 0 ❂ 1 ❀ x ❂ x ✵ or t 1 ❂ 1 ❀ y ❂ y ✵ . Intuitively, J ✭ X ❀ Y ✮ is obtained by joining each point of X with each point of Y by a unit interval. Example: J ✭ S m ❀ S n ✮ ✘ ❂ S m ✰ n ✰ 1 ✿ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Topological Join The join J ✭ X ❀ Y ✮ of two topological spaces X and Y is defined by ❂ X ✂ Y ✂ ✁ 1 ❂ ✘ ❀ def J ✭ X ❀ Y ✮ where ✭ x ❀ y ❀ t 0 ❀ t 1 ✮ ✘ ✭ x ✵ ❀ y ✵ ❀ t 0 ❀ t 1 ✮ if t 0 ❂ 1 ❀ x ❂ x ✵ or t 1 ❂ 1 ❀ y ❂ y ✵ . Intuitively, J ✭ X ❀ Y ✮ is obtained by joining each point of X with each point of Y by a unit interval. Example: J ✭ S m ❀ S n ✮ ✘ ❂ S m ✰ n ✰ 1 ✿ Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Iterated joins For p ✕ 0, the ✭ p ✰ 1 ✮ -fold join J p ✭ X ✮ of X is J p ✭ X ✮ def ✂ ✁ p ❂ ✘ ❀ ❂ X ✂ ✁ ✁ ✁ ✂ X ⑤ ④③ ⑥ ✭ p ✰ 1 ✮ times where ✭ x 0 ❀ ✿ ✿ ✿ ❀ x p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ✮ ✘ ✭ x ✵ 0 ❀ ✿ ✿ ✿ ❀ x ✵ p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ✮ i . It is easy to see that , J p ✭ S 0 ✮ , of if for each i with t i ✻ ❂ 0, x i ❂ x ✵ the zero dimensional sphere is homeomorphic to S p . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms

Introduction and some recent results Outline of the Methods Outline of the proof Connections to complexity theory Open Problems Iterated joins For p ✕ 0, the ✭ p ✰ 1 ✮ -fold join J p ✭ X ✮ of X is J p ✭ X ✮ def ✂ ✁ p ❂ ✘ ❀ ❂ X ✂ ✁ ✁ ✁ ✂ X ⑤ ④③ ⑥ ✭ p ✰ 1 ✮ times where ✭ x 0 ❀ ✿ ✿ ✿ ❀ x p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ✮ ✘ ✭ x ✵ 0 ❀ ✿ ✿ ✿ ❀ x ✵ p ❀ t 0 ❀ ✿ ✿ ✿ ❀ t p ✮ i . It is easy to see that , J p ✭ S 0 ✮ , of if for each i with t i ✻ ❂ 0, x i ❂ x ✵ the zero dimensional sphere is homeomorphic to S p . Saugata Basu Computing Betti Numbers of semi-algebraic sets – algorithms