Gluing of singular and critical points

Abstract

We suggest an approach to construction of algebraic hypersurfaces with given collection of singular points and polynomials with given collection of critical points. The approach is based on the Viro method of gluing polynomials and on the geometry of equisingular and equicritical strata in spaces of polynomials. As application we construct cuspidal plane curves of small degrees and real polynomials in two variables with given numbers of degenerate and non-degenerate critical points.