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

  title={Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials},
  author={Saugata Basu and Dmitrii V. Pasechnik and Marie-Françoise Roy},
Let R be a real closed field, Q ⊂ R[Y1, . . . , Y`, 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 ⊂ R`+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 (`smd)O(m+k). This is a common generalization of previous results in [7] and [2] on bounding the Betti numbers of closed semi-algebraic… CONTINUE READING

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