On Partial Differential Operators Which Annihilate the Roots of the Universal Equation of Degree k

Q3 Mathematics

Arnold Mathematical Journal Pub Date : 2023-06-21 DOI:10.1007/s40598-023-00229-2

Daniel Barlet

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引用次数: 0

Abstract

The aim of this paper is to study in details the regular holonomic $D-$module introduced in Barlet (Math Z 302 $n^03$: 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface $\{\Delta (\sigma ).\sigma _k = 0 \}$ are given by the local system generated by the power $\lambda $ of the local branches of the multivalued function which is the root of the universal degree k equation $z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0 $. We show that for $\lambda \in \mathbb {C} {\setminus } \mathbb {Z}$ this D-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set $\{\sigma _k\Delta (\sigma ) \not = 0\}$. We then study the structure of these D-modules in the cases where $\lambda = 0, 1, -1$ which are a little more complicated, but which are sufficient to determine the structure of all these D-modules when $\lambda $ is in $\mathbb {Z}$. As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near $-1$ of the equation:

$$\begin{aligned} z^k + \sum _{h=-1}^k (-1)^h\sigma _hz^{k-h} - (-1)^k = 0. \end{aligned}$$

near $z^k - (-1)^k = 0$.

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关于消去k次通用方程根的偏微分算子

本文旨在详细研究 Barlet (Math Z 302 $n^03$：1627-1655, 2022 arXiv:1911.09347 [math]）中引入的正则全局模块，其极性超曲面外的局部解是 \{\Delta (\sigma ).$z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0 $的根的多值函数的局部分支的幂$\lambda $产生的局部系统给出。）我们证明了对于（(\lambda \in \mathbb {C} {\setminus } \mathbb {Z}/)）这个 D 模块是全纯向量束的最小扩展，它有一个可积分的全纯连接，这个连接有一个简单极点，是它在开集 $\{\sigma _k\Delta (\sigma ) \not = 0\}$ 上的限制。然后我们研究这些D模块在$\lambda = 0, 1, -1$ 的情况下的结构，这些情况稍微复杂一些，但是当$\lambda $在$\mathbb {Z}$中时，它们足以决定所有这些D模块的结构。作为一个应用，我们展示了这些结果如何允许计算，例如，在方程的$-1$附近根的泰勒展开：$$\begin{aligned} z^k + \sum _{h=-1}^k (-1)^h\sigma _hz^{k-h} - (-1)^k = 0. \end{aligned}$$$z^k-(-1)^k=0$附近。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.