论文标题
$ \ MATHCAL C^M $ SEMIALGEBRAIC或可定义方程式的解决方案
$\mathcal C^m$ solutions of semialgebraic or definable equations
论文作者
论文摘要
我们解决了以下问题：（1）惠特尼扩展问题中的解决方案以及（2）Brenner-Fefferman-Hochster-Kollár问题是否可以保留给定数据上的几何条件，这两个问题都适用于$ \ MATHCAL C^M $函数。我们的结果涉及一定的可怜性丧失。 问题（2）涉及线性方程系统$ a（x）g（x）= f（x）$的解决方案，其中$ a $是$ \ mathbb r^n $上的函数矩阵，而$ f $，$ g $是vector值函数。假设$ a（x）$的条目是半gebraic（或更一般地，在合适的O小型结构中可以定义）。然后，我们找到$ r = r（m）$，这样，如果$ f（x）$是可定义的，并且系统接纳了$ \ MATHCAL C^r $解决方案$ g（x）$，则有一个$ \ MATHCAL C^M $可定义的解决方案。同样，在问题（1）中，给定可封闭的可定义子集$ x $的$ \ m artbb r^n $，我们发现$ r = r（m）$，这样，如果$ g：x \ to \ mathbb r $可定义，并且扩展到$ \ mathcal c^r $在$ \ mathbb r r^n $上的$ \ mathbb r^n $，则是$ \ mathable $ \ $ \ $ $ \ $ $ $ $ $ c^m $ dex c^m n $ decte。
We address the question of whether geometric conditions on the given data can be preserved by a solution in (1) the Whitney extension problem, and (2) the Brenner-Fefferman-Hochster-Kollár problem, both for $\mathcal C^m$ functions. Our results involve a certain loss of differentiability. Problem (2) concerns the solution of a system of linear equations $A(x)G(x)=F(x)$, where $A$ is a matrix of functions on $\mathbb R^n$, and $F$, $G$ are vector-valued functions. Suppose the entries of $A(x)$ are semialgebraic (or, more generally, definable in a suitable o-minimal structure). Then we find $r=r(m)$ such that, if $F(x)$ is definable and the system admits a $\mathcal C^r$ solution $G(x)$, then there is a $\mathcal C^m$ definable solution. Likewise in problem (1), given a closed definable subset $X$ of $\mathbb R^n$, we find $r=r(m)$ such that if $g:X\to\mathbb R$ is definable and extends to a $\mathcal C^r$ function on $\mathbb R^n$, then there is a $\mathcal C^m$ definable extension.
