Spaces of polynomials with real roots of bounded multiplicity

We describe an alternative approach to some results of Vassiliev ([Va1]) on spaces of polynomials, by using the “scanning method” which was used by Segal ([Se2]) in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.