Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

Abstract

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in ${\mathbb{R}}^{\ell }$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\mathbb{R}$ be a real closed field and let \[ {\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], \] with ${\mathrm{deg}}_Y(P_i)\le 2,{\mathrm{deg}}_X(P_i)\le d,1\le i\le m$. Let $S\subset {\mathbb{R}}^{\ell +k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P\ge 0,P\le 0,P\in \mathcal{P}$. Let $\pi :{\mathbb{R}}^{\ell +k}\to {\mathbb{R}}^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers $S_{\x} = π^{-1}(\x) \cap S$ is bounded by \[ (2^m\ell k d)^{O(mk)}. \]

27 pages, 1 figure

References in corpus (1)

• On the number of topological types occurring in a parametrized family of arrangements

Cited by in corpus (1)

• A complexity theory of constructible functions and sheaves