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 τ. Our bound is an explicit
function of s,d,k and τ, 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 2τdO(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 τ. The bounds
proved in this paper are of this nature.Comment: 11 page