Some Quantitative Results in Real Algebraic Geometry

Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry, primarily upper bounds on the topological complexity of semi-algebraic sets as measured, for example, by their Betti numbers. Another quantitative measure of topological complexity which we study is the number of homotopy types of semi-algebraic sets of bounded description complexity. The description complexity of a semi-algebraic set depends on the context, but it is simply some measure of the complexity of the polynomials in the formula defining the semi-algebraic set (e.g., the degree and number of variables of a polynomial, the so called dense format). We also provide a description of the Hausdorff limit of a one-parameter family of semi-algebraic sets, up to homotopy type which the author hopes will find further use in the future. Finally, in the last chapter of this thesis, we prove a decomposition theorem similar to the well-known cylindrical decomposition for semi-algebraic sets but which has the advantage of requiring significantly less cells in the decomposition. The setting of this last result is not in semi-algebraic geometry, but in the more general setting of o-minimal structures.