Bounding the Betti numbers and computing the Euler–Poincaré characteristic of semi-algebraic sets deﬁned by partly quadratic systems of polynomials
Pasechnik, Dmitrii V.
Date of Issue2010
School of Physical and Mathematical Sciences
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)).
Journal of the European mathematical society
© 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].