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

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 $S\subset \text{mathbbm}{R}^k$ defined by a quantifier-free formula involving $s$ polynomials in $\text{mathbbm}Z[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_k]$ 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.