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

We prove an explicit bound 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 of R(k) defined by a weak sign condition involving s polynomials in 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 d0(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. (C) 2010 Elsevier Ltd. All rights reserved.