Real hypersurfaces with many simple singularities (Q2574715)

This paper extends the results of \textit{E. Shustin} and \textit{E. Westenberger} [J. Lond. Math. Soc., II. Ser. 70, No. 3, 609--624 (2004; Zbl 1075.14034)] on the existence of algebraic hypersurfaces with prescribed simple singularities to the real case. The main result states that, given a collection of real simple singularities (with repetitions) of finitely many types satisfies the condition that the total Milnor number does not exceed \(d^n/n!+O(d^{n-1})\), there exists a real projective \((n-1)\)-dimensional hypersurface of degree \(d\), whose real singularity set coincides with the given collection, and, furthermore, the germ of the corresponding equisingular stratum in the space of hypersurfaces of degree \(d\) is smooth of expected dimension. The proof reduces to an application of the patchworking construction along the lines of the above cited article.