关于次解析函数的局部极值集
On the set of local extrema of a subanalytic function
摘要：
设${\mathfrak F}$是实解析流形的子解析子集的一个范畴，它在基本集合论和基本拓扑运算下是闭的。设$M$是一个实解析流形，用${\mathfrak F}(M)$表示属于${\mathfrak F}$的$M$的子集族.设$f:X\to{\mathbb R}$是子集$X\in{\mathfrak F}(M)$上的子解析函数，使得${\mathbb R}$的每个区间在$f$下的逆像属于${\mathfrak F}(M)$。令${\rm Max}(f)$是$f$的局部最大值的集合，并考虑每个$λ\in{\mathbb R}$的${\rm Max}_λ(f):={\rm Max}(f)\cap\{f=λ\}$。如果$f$是连续的，则${\rm Max}(f)=\bigsqcup_{λ\in{\mathbb R}}{\rm Max}_λ(f)\in{\mathfrak F}(M)$当且仅当族$\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$在$M$中是局部有限的。如果去掉连续性条件，则存在次解析函数$f:X\to M$，使得${\rm Max}(f)\in{\mathfrak F}(M)$，但族$\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$在$M$中不是局部有限的，或使得${\rm Max}(f)$是连通的，但它甚至不是次解析的.如果${\mathfrak F}$是子解析集范畴，$f:X\to{\mathbb R}$是一个子解析映射$f$，它将包含在$X$中的$M$的相对紧子集映射到${\mathbb R}$的有界子集，则${\rm Max}(f)\in{\mathfrak F}(M)$和族$\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$在$M$中是局部有限的。若范畴${\mathfrak F}$包含代数集与实解析子流形的交，且$X\in{\mathfrak F}(M)$在$M$中不闭，则存在一个连续的子解析函数$f:X\to{\mathbb R}$，其图属于${\mathfrak F}(M\times{\mathbb R})$，使得${\mathbb R}$的区间在$f$下的逆像属于${\mathfrak F}(M)$，而${\rm Max}(f)$不属于${\mathfrak F}(M)$ .
英文摘要：
Let ${\mathfrak F}$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical and basic topological operations. Let $M$ be a real analytic manifold and denote ${\mathfrak F}(M)$ the family of the subsets of $M$ that belong to ${\mathfrak F}$. Let $f:X\to{\mathbb R}$ be a subanalytic function on a subset $X\in{\mathfrak F}(M)$ such that the inverse image under $f$ of each interval of ${\mathbb R}$ belongs to ${\mathfrak F}(M)$. Let ${\rm Max}(f)$ be the set of local maxima of $f$ and consider ${\rm Max}_λ(f):={\rm Max}(f)\cap\{f=λ\}$ for each $λ\in{\mathbb R}$. If $f$ is continuous, then ${\rm Max}(f)=\bigsqcup_{λ\in{\mathbb R}}{\rm Max}_λ(f)\in{\mathfrak F}(M)$ if and only if the family $\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$ is locally finite in $M$. If we erase continuity condition, there exist subanalytic functions $f:X\to M$ such that ${\rm Max}(f)\in{\mathfrak F}(M)$, but the family $\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$ is not locally finite in $M$ or such that ${\rm Max}(f)$ is connected but it is not even subanalytic. If ${\mathfrak F}$ is the category of subanalytic sets and $f:X\to{\mathbb R}$ is a subanalytic map $f$ that maps relatively compact subsets of $M$ contained in $X$ to bounded subsets of ${\mathbb R}$, then ${\rm Max}(f)\in{\mathfrak F}(M)$ and the family $\{{\rm Max}_λ(f)\}_{λ\in{\mathbb R}}$ is locally finite in $M$. If the category ${\mathfrak F}$ contains the intersections of algebraic sets with real analytic submanifolds and $X\in{\mathfrak F}(M)$ is not closed in $M$, there exists a continuous subanalytic function $f:X\to{\mathbb R}$ with graph belonging to ${\mathfrak F}(M\times{\mathbb R})$ such that inverse images under $f$ of the intervals of ${\mathbb R}$ belong to ${\mathfrak F}(M)$ but ${\rm Max}(f)$ does not belong to ${\mathfrak F}(M)$.
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