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Let $\R$ be a real closed field, $ {\mathcal Q} \subset\R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leqd, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and $ {\mathcal P} \subset\R[X_1,...,X_k] $ with $\deg_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcalP})=s$, and $S \subset \R^{\ell+k}$ a semi-algebraic set defined by a Booleanformula without negations, with atoms $P=0, P \geq 0, P \leq 0, P \in {\mathcalP} \cup {\mathcal Q}$. We prove that the sum of the Betti numbers of $S$ isbounded by \[ \ell^2 (O(s+\ell+m)\ell d)^{k+2m}. \] This is a commongeneralization of previous results on bounding the Betti numbers of closedsemi-algebraic sets defined by polynomials of degree $d$ and 2, respectively.  We also describe an algorithm for computing the Euler-Poincar\'echaracteristic of such sets, generalizing similar algorithms known before. Thecomplexity of the algorithm is bounded by $(\ell s m d)^{O(m(m+k))}$.

Bounding the Betti numbers and computing the Euler–Poincaré characteristic of semi-algebraic sets deﬁned by partly quadratic systems of polynomials