Bounding the radii of balls meeting every connected component of semi-algebraic sets

TL;DR

This paper establishes explicit bounds on the radius of a ball centered at the origin that contains or intersects all connected components of semi-algebraic sets, with bounds depending explicitly on defining parameters.

Contribution

It provides explicit, parameter-dependent bounds on the radii of balls intersecting or containing all connected components of semi-algebraic sets, improving upon previous asymptotic bounds.

Findings

Explicit bounds depend on s, d, k, τ

Bounds apply to both bounded and unbounded components

No undetermined constants in the bounds

Abstract

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$ polynomials in $\mathbbm{Z}[X_1, ..., X_k]$ having degrees at most $d$, and whose coefficients have bitsizes at most $\tau$. Our bound is an explicit function of $s, d, k$ and $\tau$, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of $S$ (including the unbounded components). While asymptotic bounds of the form $2^{\tau d^{O (k)}}$ on these quantities were known before, some applications require bounds which are explicit and which hold for all values of $s, d, k$ and $\tau$. The bounds proved in this paper are of this nature.