Bipolar varieties and real solving of a singular polynomial equation (Q2895243)

The problem is treated in finding real solutions for a polynomial equation, i.e., more precisely in the language of geometry, to find algebraic sample points for the connected components of a singular real hypersurface. Algorithms are designed and complexity bounds are given. The complexity of the algorithms is polynomial in the maximal geometric degree of the so-called bipolar varieties of the given hypersurface. An explicit example is treated. The results can be viewed as a local version of the complexity statements of section 5 of the paper of the authors together with \textit{L. Lehmann} [``Algorithms of intrinsic complexity for point searching in compact real singular hypersurfaces'', Found. Comput. Math. 12, No. 1, 75--122 (2012; Zbl 1246.14071)].