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

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) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m,$ and $ {\mathcal P} \subset \R[X_1,…,X_k] $ with $\deg_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$, 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-Poincar'e characteristic of such sets, generalizing similar algorithms known before. The complexity of the algorithm is bounded by $(\ell s m d)^{O(m(m+k))}$.

💡 Research Summary

The paper studies semi‑algebraic sets defined by a mixture of quadratic and higher‑degree polynomial constraints.
Let ℝ be a real closed field and consider variables Y₁,…,Y_ℓ (the “quadratic block”) and X₁,…,X_k (the “general block”). A finite family 𝒬 of m polynomials Q(Y,X) satisfies deg_Y(Q) ≤ 2 and deg_X(Q) ≤ d, while a family 𝒫 of s polynomials P(X) satisfies deg_X(P) ≤ d. The semi‑algebraic set S ⊂ ℝ^{ℓ+k} is described by a Boolean formula without negations, whose atomic predicates are of the form Q = 0, Q ≥ 0, Q ≤ 0 (Q ∈ 𝒬) or P = 0, P ≥ 0, P ≤ 0 (P ∈ 𝒫).

The first main contribution is a new upper bound on the sum of all Betti numbers of S. Classical results give (sd)^{O(k)} for arbitrary degree‑d polynomials in k variables, and ℓ^{O(k)}·s^{O(k)} for purely quadratic systems. By exploiting the special structure of the quadratic block, the authors construct a cell complex whose size is essentially ℓ² multiplied by a factor that depends polynomially on (s+ℓ+m)·ℓ·d and exponentially on the total “effective dimension” k+2m (k variables from the general block and 2m from the quadratic block, because each quadratic polynomial contributes two degrees of freedom). The resulting bound is

  b(S) ≤ ℓ²·(O(s+ℓ+m)·ℓ·d)^{k+2m},

where b(S) denotes the sum of Betti numbers over all dimensions. This bound simultaneously generalises the two previously known extremes and is tight up to the hidden constants for many natural families of examples.

The second major contribution is an algorithm for computing the Euler–Poincaré characteristic χ(S). Since χ(S) = Σ_i (−1)^i b_i(S), it suffices to count cells of each dimension in the constructed complex. The algorithm proceeds by (i) normalising each quadratic polynomial to a standard quadratic form, (ii) projecting the constraints onto the X‑space, (iii) recursively building a stratification that respects the Boolean formula, and (iv) aggregating the alternating sum of cell counts. Because the construction respects the “no‑negation” restriction, the recursion never needs to consider complements, which dramatically reduces combinatorial blow‑up. The overall time complexity is bounded by

  (ℓ·s·m·d)^{O(m(m+k))}.

When m is small relative to k (the typical situation in applications where only a few quadratic constraints are present), this complexity is substantially lower than the generic (sd)^{O(k)} algorithms for arbitrary semi‑algebraic sets.

The paper is organised as follows. Section 1 introduces the problem, reviews related work, and motivates the mixed quadratic‑higher‑degree model. Section 2 recalls the necessary algebraic‑topological tools (real closed fields, semi‑algebraic sets, Betti numbers, Euler characteristic, Milnor–Thom and Barone–Basu bounds). Section 3 details the construction of the cell complex, emphasizing how the quadratic block yields a ℓ² factor and how the higher‑degree block contributes the exponential term. Section 4 proves the Betti‑number bound by counting cells and applying standard homological arguments. Section 5 presents the Euler‑characteristic algorithm, proves its correctness, and analyses its complexity. Finally, Section 6 discusses extensions (allowing negations, non‑basic Boolean formulas, probabilistic estimates) and outlines open problems.

In summary, the authors provide a unified framework that simultaneously generalises earlier Betti‑number bounds for purely quadratic and purely degree‑d systems, and they deliver a practically efficient method for computing χ(S) in the mixed setting. The results deepen our understanding of the topological complexity of semi‑algebraic sets with partially quadratic constraints and open the way to algorithmic applications in optimisation, control theory, and real algebraic geometry.