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单词 CartansUmbrella
释义

Cartan’s umbrella


The Cartan’s umbrella refers to a certain class of examples of real analytic sets (in fact real algebraic usually) in 3,which are irreducible (not written as a union of proper subsetsMathworldPlanetmathPlanetmath that are also subvarietiesMathworldPlanetmath), and wherethere are regular points both of dimensionMathworldPlanetmath 1 and of dimension 2. Sometimes higher dimensional examples with similar behavior are also called the same. A fairly common equation for a Cartan umbrella is z(x2+y2)-y3=0. Solving for zwe get z=y3x2+y2. The graph of this function is shown in the following figure.


Figure 1: Graph of z=y3x2+y2

The umbrella itself also includes the z axis, since all points of the form (0,0,z) satisfy the equation. . It is impossible to write an equation (real analytic or real algebraic) whose solution set contains the graph in Figure 1, and such that the z axis is not included.

This pathological behavior does not happen for complex analytic subvarieties.

References

  • 1 Jacek Bochnak, Michel Coste, Marie-Francoise Roy..Springer, 1998.
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更新时间:2025/5/4 3:02:17