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

TL;DR

This paper establishes a nearly optimal exponential bound in the number of parameters and inequalities on the stable homotopy types of semi-algebraic sets defined by quadratic inequalities, with the bound polynomial in the ambient dimension.

Contribution

It provides a new bound on the number of stable homotopy types for parametrized semi-algebraic sets defined by quadratic inequalities, improving understanding of their topological complexity.

Findings

Bound is exponential in parameters and inequalities, polynomial in ambient dimension.

Number of stable homotopy types is at most (2^m * l * k * d)^{O(mk)}.

Result applies to semi-algebraic sets defined by quadratic inequalities in real closed fields.

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 $\R^\ell$, 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 $\R$ be a real closed field and let \[ {\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], \] with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \geq 0, P\leq 0, P \in {\mathcal P}$. Let $\pi: \R^{\ell+k} \to \R^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers $S_{\x} = \pi^{-1}(\x) \cap S$ is bounded by \[ (2^m\ell k d)^{O(mk)}. \]