Bounding the Betti numbers and computing the Euler-Poincar\'e characteristic of semi-algebraic sets defined by partly quadratic systems of polynomials

TL;DR

This paper establishes bounds on the Betti numbers and provides an algorithm for computing the Euler-Poincaré characteristic of semi-algebraic sets defined by partly quadratic polynomial systems, generalizing previous results.

Contribution

It introduces new bounds for Betti numbers and an algorithm for Euler-Poincaré characteristic computation for semi-algebraic sets with partly quadratic polynomials.

Findings

Betti number sum bounded by ilde{ ext{bound}}

Algorithm for Euler-Poincaré characteristic with complexity ilde{ ext{complexity}}

Generalizes previous bounds and algorithms for quadratic and degree-d polynomials.

Abstract

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 $(\ell s m d)^{O(m(m+k))}$.