We give some background and describe the very recent result in quantitative real algebraic geometry of B.-Basu on bounding the sum of the Betti numbers of sign conditions on a variety, in terms of the number, degree, and number of variables of the polynomials involved. The classical results are due to Oleinik-Petrovsky (1949), Thom (1965), and Milnor (1964), and also to Warren (1968). More recent results of Basu-Pollack-Roy (2005), and Gabrielov-Vorobjov (2005) extended these to include a larger class of semi-algebraic sets. In the last few years, B.-Basu (2012 and 2013) refined the bounds, on the number of connected components only, to include separate parameters for the degrees of the polynomials involved. We prove that the sum of the Betti numbers of sign conditions on a variety of dimension k' is at most (sd)^{k'} d_0^{O(kk')}, where d is a bound on the degrees of the polynomials defining the sign conditions and d_0 a bound on those defining the variety. This can be seen as a weak extension of the recent results of B.-Basu and a weak refinement of the earlier results taking into account, for the first time, the different roles played by the polynomials defining the sign conditions and those defining the variety in a bound on the higher Betti numbers.
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