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|Title:||Bounding the Betti numbers and computing the Euler–Poincaré characteristic of semi-algebraic sets deﬁned by partly quadratic systems of polynomials||Authors:||Basu, Saugata
Pasechnik, Dmitrii V.
|Keywords:||DRNTU::Science::Mathematics::Algebra||Issue Date:||2010||Source:||Basu, S., Pasechnik, D. V., & Roy, M.- F. Bounding the Betti numbers and computing the Euler–Poincaré characteristic of semi-algebraic sets deﬁned by partly quadratic systems of polynomials. Journal of the European Mathematical Society, 12(2), 529-553.||Series/Report no.:||Journal of the European mathematical society||Abstract:||Let R be a real closed ﬁeld, Q ⊂ R[Y1 , . . . , Yl, X1 , . . . , Xk], with degY(Q) ≤ 2, degX(Q) ≤ d, Q ∈ Q, #(Q) = m, and P ⊂ R[X1, . . . , Xk] with degX(P) ≤ d, P ∈ P, #(P) = s, and S ⊂ Rl+k a semi-algebraic set deﬁned by a Boolean formula without negations, with atoms P = 0, P ≥ 0, P ≤ 0, P ∈ P ∪ Q. We prove that the sum of the Betti numbers of S is bounded by l2 (O(s + l + m)ld)k+2m. This is a common generalization of previous results in  and  on bounding the Betti numbers of closed semi-algebraic sets deﬁned by polynomials of degree d and 2, respectively. We also describe an algorithm for computing the Euler–Poincaré characteristic of such sets, e generalizing similar algorithms described in [4, 9]. The complexity of the algorithm is bounded by (lsmd)O(m(m+k)).||URI:||https://hdl.handle.net/10356/98233
|ISSN:||1435-9855||DOI:||10.4171/JEMS/208||Rights:||© 2010 European Mathematical Society. This is the author created version of a work that has been peer reviewed and accepted for publication by Journal of the European Mathematical Society, European Mathematical Society. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: DOI[http://dx.doi.org/10.4171/JEMS/208].||Fulltext Permission:||open||Fulltext Availability:||With Fulltext|
|Appears in Collections:||SPMS Journal Articles|
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