论文标题
功能射流理想的属性消失了
A property of ideals of jets of functions vanishing on a set
论文作者
论文摘要
对于包含我们考虑的来源的$ e \ subset \ mathbb {r}^n $，我们考虑的$ i^m（e）$ - 所有$ m^{\ text {t text {th}} $ teg taylor近似（在$ c^m $ the $ e $ e $ $ e $上的$ c^m $函数的$ c^m $函数）的集合。该集合是$ \ Mathcal {p}^m（\ Mathbb {r}^n）$的理想选择 - 所有$ m^{\ text {tht}} $ c^m $ taylor近似$ c^m $ functions $ \ \ \ \ \ mthbb {r}^n $ functions的环。 $ \ Mathcal {p}^m（\ Mathbb {r}^n）$中的哪些理想为某些$ e $？在本文中，我们介绍了$ \ mathcal {p}^m（\ mathbb {r}^n）$中的一个\ textit {note}理想的概念，并证明了$ i^m（e）$的任何理想是关闭的。我们不知道某些$ e $的$ i^m（e）$一般而言是否是任何封闭的理想，但是我们在[fs]中证明，所有封闭的理想中的所有封闭的理想都以$ \ mathcal {p}^m（\ mathbb {r}^n）$作为$ i^m（e）$时，当$ m+n \ leq5 $。
For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish on $E$. This set is an ideal in $\mathcal{P}^m(\mathbb{R}^n)$ -- the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. Which ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ for some $E$? In this paper we introduce the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and prove that any ideal of the form $I^m(E)$ is closed. We do not know whether in general any closed ideal is of the form $I^m(E)$ for some $E$, however we prove in [FS] that all closed ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$.
