Title:Some New Bounds in Subanalytic Geometry
Abstract: Suppose you'd like to know how many real connected components are defined by the set of solutions to a given system of analytic inequalities. Thanks to work of Oleinik, Petrovsky, Milnor, and Thom starting around 1949, we at least now know an upper bound in the special case of polynomials: the bound is a function of the polynomial degrees and is exponential in the number of variables.
Thanks to work of Benedetti, Risler, and Loeser (around 1991), there is a better bound stated in terms of Newton polytopes which can be sharper than the previous bound by a factor exponential in the number of variables. Furthemore, a bit earlier, Khovanski a bound which depends solely on the number monomial terms, and extends to real exponents.
We show how to improve all these bound in a simple way. The underlying tricks come down to picking clever projections of the underlying set and counting critical points. No background in semi-algebraic geometry is assumed.