论文标题
在某些喷气机中的含义关联理想的分类
Classification of implication-closed ideals in certain rings of jets
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 $函数）的集合。该集合在$ \ Mathcal {p}^m（\ mathbb {r}^n）$中是一个适当的理想 - 所有$ m^{\ text {tht}} $ c^m $函数的所有$ m^{\ text {tht}} $ taylor近似值。在[fs]中，我们介绍了$ \ mathcal {p}^m（\ mathbb {r}^n）$中的一个\ textit {note}的概念，并证明了$ i^m（e）$的任何理想是关闭的。在本文中，我们将所有封闭的理想分类为$ \ mathcal {p}^m（\ mathbb {r}^n）$在$ m+n \ leq5 $的所有情况下。我们还表明，在这种情况下，相反的情况也可以 - 所有封闭的适当理想都以$ \ MATHCAL {p}^m（\ MATHBB {r}^n）$呈现为$ i^m（e）$时，当$ m+n \ leq5 $。此外，我们证明在这些情况下，对于某些$ i^m（e）$的任何理想，对于某些$ e \ subset \ subset \ mathbb {r}^n $，其中包含该来源的$ i^m（v）$对于某些半级别$ v \ subset $ subset \ subset \ subset \ mthbb {r}^n $，其中包含起源。通过这样做，我们证明了N. Zobin的猜想在这些情况下是正确的。
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 a proper ideal in $\mathcal{P}^m(\mathbb{R}^n)$ -- the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. In [FS] we introduced the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and proved that any ideal of the form $I^m(E)$ is closed. In this paper we classify (up to a natural equivalence relation) all closed ideals in $\mathcal{P}^m(\mathbb{R}^n)$ in all cases in which $m+n\leq5$. We also show that in these cases the converse also holds -- all closed proper ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$. In addition, we prove that in these cases any ideal of the form $I^m(E)$ for some $E\subset\mathbb{R}^n$ that contains the origin already arises as $I^m(V)$ for some semi-algebraic $V\subset\mathbb{R}^n$ that contains the origin. By doing so we prove that a conjecture by N. Zobin holds true in these cases.
