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 ⊂ Rk defined by a quantifier-free formula involving s polynomials in Z[X1, . . . ,Xk] 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 2 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 τ . The bounds proved in this paper are of this nature.