设$\R$是实闭域，$ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $有$ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$，$ {\mathcal P} \subset \R[X_1,...,X_k] $有$°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$，$S \subset \R^{\ell+k}$是由布尔公式定义的半代数集，无否定，有原子$P=0, P \geq 0, P \leq 0, P \in {\mathcal P} \cup {\mathcal Q}$ .证明了$S$的贝蒂数之和是由\[ \ell^2 (O(s+\ell+m)\ell d)^{k+2m}. \]有界的。这是以前关于分别由$d$和2次多项式定义的闭半代数集的贝蒂数的有界结果的一般推广。我们还描述了一个算法，用于计算欧拉-庞加莱特性的这样的集合，推广类似的算法之前已知的。该算法的复杂度以$(\ell s m d)^{O(m(m+k))}$为界。
英文摘要：
Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, and $S \subset \R^{\ell+k}$ a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \geq 0, P \leq 0, P \in {\mathcal P} \cup {\mathcal Q}$. We prove that the sum of the Betti numbers of $S$ is bounded by \[ \ell^2 (O(s+\ell+m)\ell d)^{k+2m}. \] This is a common generalization of previous results 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, generalizing similar algorithms known before. The complexity of the algorithm is bounded by $(\ell s m d)^{O(m(m+k))}$.
