Please use this identifier to cite or link to this item: https://hdl.handle.net/10356/98233
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
Roy, Marie-Françoise
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 [4] and [3] 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
http://hdl.handle.net/10220/9277
ISSN: 1435-9855
DOI: http://dx.doi.org/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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