Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials

• Main Authors: Basu, Saugata, Pasechnik, Dmitrii V., Roy, Marie-Françoise
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2009
• Subjects: DRNTU::Science::Mathematics::Geometry
• Online Access: https://hdl.handle.net/10356/101279; http://hdl.handle.net/10220/4599; http://sfxna09.hosted.exlibrisgroup.com:3410/ntu/sfxlcl3?sid=metalib:ELSEVIER_SCOPUS&id=doi:&genre=&isbn=&issn=&date=2009&volume=321&issue=8&spage=2206&epage=2229&aulast=Basu&aufirst=%20S&auinit=&title=Journal%20of%20Algebra&atitle=Computing%20the%20Betti%20numbers%20of%20semi%2Dalgebraic%20sets%20defined%20by%20partly%20quadratic%20systems%20of%20polynomials
• Institution: Nanyang Technological University

Let R be a real closed field, , with degY(Q)2, degX(Q)d, , , and with degX(P)d, , . Let SRℓ+k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P=0, P0, P0, . We describe an algorithm for computing the Betti numbers of S generalizing a similar algorithm described in [S. Basu, Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (1) (2008) 45–80]. The complexity of the algorithm is bounded by ((ℓ+1)(s+1)(m+1)(d+1))2O(m+k). The complexity of the algorithm interpolates between the doubly exponential time bounds for the known algorithms in the general case, and the polynomial complexity in case of semi-algebraic sets defined by few quadratic inequalities [S. Basu, Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (1) (2008) 45–80]. Moreover, for fixed m and k this algorithm has polynomial time complexity in the remaining parameters.