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

@article{Basu2007BettiNO,
  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},
  journal={CoRR},
  year={2007},
  volume={abs/0708.3522}
}
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

From This Paper

Figures, tables, and topics from this paper.

References

Publications referenced by this paper.
Showing 1-10 of 32 references

Topology of quadratic maps and Hessians of smooth maps

A. A. Agrachev
Algebra, Topology, Geometry, Vol 26 (Russian),85-124, 162, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn.i Tekhn. Inform., Moscow, 1988. Translated in J. Soviet Mathematics. 49 • 1990
View 6 Excerpts
Highly Influenced

Sur l’homologie des variétés algébriques réelles

R. Thom
Differential and Combinatorial Topology, 255–265. Princeton University Press, Princeton • 1965
View 4 Excerpts
Highly Influenced

Efficient algorithm for computing the Euler-Poincaré characteristic of semialgebraic sets defined by few quadratic inequalities

S. Basu
Computational Complexity, 15 • 2006
View 3 Excerpts

Similar Papers

Loading similar papers…