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Real algebraic varieties with prescribed tangent cones. (English) Zbl 1036.14027
From the introduction: The definition of the tangent cone \(C(V,p)\) to an algebraic variety \(V\) at a point \(p\) was given by Whitney more than 40 years ago as one of the tools to get information about the geometric shape of a variety near a singular point. While the complex case has been widely and successfully studied, including from a computational point of view, only recently have some first attempts been made to elucidate the situation in the real case. Since the tangent cone to a real algebraic variety is a semialgebraic set, a question is that of investigating which semialgebraic cones of \(\mathbb{R}^n\) can be realized as tangent cones to real algebraic subsets of \(\mathbb{R}^n\). In this paper we show that any closed semialgebraic semicone (i.e., a union of rays) of codimension at least one in \(\mathbb{R}^n\) is the tangent semicone (i.e., a union of limits of secant rays) to a suitable real algebraic variety in \(\mathbb{R}^n\).

14P25 Topology of real algebraic varieties
14P05 Real algebraic sets
14P10 Semialgebraic sets and related spaces
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