Geodesic
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, Tome 12 (2010) no. 2, pp. 529-553.

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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
DOI : 10.4171/jems/208
Classification : 14-XX, 68-XX, 00-XX
Keywords: Betti numbers, quadratic inequalities, semi-algebraic sets 
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     title = {Bounding the {Betti} numbers and computing the {Euler{\textendash}Poincar\'e} characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials},
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Saugata Basu; Dmitrii V. Pasechnik; Marie-Françoise Roy. 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, Tome 12 (2010) no. 2, pp. 529-553. doi : 10.4171/jems/208. http://geodesic.mathdoc.fr/articles/10.4171/jems/208/

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