  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.
