关于o-极小结构中可定义函数的分支集
On the bifurcation sets of functions definable in o-minimal structures
摘要：
设g：X -> Y是两个光滑流形之间的光滑（即C^ \infty可微）映射.与复多项式函数的情况类似，我们说Y中的y_0是g的典型值，如果Y中存在y_0的开邻域U，使得限制g：g^{-1}（U）-> U是C^ \infty平凡纤维化。如果Y中的y_0不是g的典型值，则y_0称为g的非典型值。我们用B_g表示g的分歧集，即g的非典型值集。在复多项式函数f：C^n -> C的情况下，已知B_f是有限集。以前证明了实多项式函数的分支集也是有限的。本文的目的是证明光滑可定义函数f：R^n -> R的分支集B_f是有限的。
英文摘要：
Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two smooth manifolds. In analogy with the case of complex polynomial functions, we say that y_0 in Y is a typical value of g if there exists an open neighbourhood U of y_0 in Y, such that the restriction g:g^{-1}(U) -> U is a C^\infty trivial fibration. If y_0 in Y is not a typical value of g, then y_0 is called an atypical value of g. We denote by B_g the bifurcation set of g, i.e. the set of atypical values of g. In the case of a complex polynomial function f:C^n -> C it is known that B_f is a finite set. It was previously proved that the bifurcation sets of real polynomial functions are also finite. The aim of this note is to show that the bifurcation set B_f of a smooth definable function f:R^n -> R is finite .
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