Mathematical Sciences: Model Theory and Algebra

数学科学：模型理论和代数

• 批准号: 8802410
• 负责人: Leonard Lipshitz
• 金额: $ 11.11万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-15 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8802410&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Model; Theory; Algebra

This project continues the development of a theory for treating decidability questions about analytic solutions of systems of differential equations. For example, the principal investigator and J. Denef have previously shown that there is no algorithm for deciding if a single ordinary differential equation has an analytic solution on the non-negative real numbers, even given that it has a unique solution near zero. Part 1 of the project is the development of a theory of nonarchimedean rigid semi-analytic and sub-analytic sets. (By this is meant the theory of semi-analytic and sub-analytic sets over a nonarchimedean, algebraically closed, complete normed field.) This development will parallel the recent treatment of p-adic semi-analytic and sub-analytic sets of Denef and van den Dries. The principal investigator has proved an analytic elimination theorem (over nonarchimedean, algebraically closed, complete normed fields) which will facilitate the development of the theory. Results sought include (a) that the complement of a rigid sub-analytic set is subanalytic; (b) that every subanalytic set is a finite disjoint union of subanalytic manifolds; (c) nonarchimedean Lojasiewicz inequalities for rigid subanalytic sets and functions; and some further results. In part 2 the principal investigator seeks to show that algebraic subsets of the p-adic integers Zp, defined by polynomials involving few monomials, if they are finite contain only few points in Zp. In part 3, he will investigate a decision problem about the existence of power series solutions of maximally overdetermined systems of partial differential equations.

该项目继续开发一种理论，用于处理有关微分方程系统分析解决方案的可确定性问题。 例如，主要研究者和J. DENEF先前已经表明，没有算法来确定单个普通微分方程在非负实数上是否具有分析解决方案，即使它在零接近零方面具有独特的解决方案。 该项目的第1部分是开发非固定刚性半分析和子分​​析集的理论。 （这是指在非构造的，代数封闭的，完整的规范领域上的半分析和子分​​析集的理论。）这种发展将与最近对P-ADIC半分析和亚分析集的处理相似。 首席研究者已证明了一种分析性消除定理（超过构造的，代数封闭，完整的规范领域），这将有助于理论的发展。 寻求的结果包括（a）刚性亚分析集的补体是亚分析的； （b）每个亚分析集都是亚分析歧管的有限分离； （c）刚性亚分析集和功能的非一切制lojasiewicz的不平等；还有一些进一步的结果。 在第2部分中，主要研究人员试图表明，如果有限的ZP中只有很少的点，则P-Adic整数ZP的代数子集，该ZP的定义很少。 在第3部分中，他将研究一个决策问题，内容涉及偏微分方程最大化系统的功率系列解决方案的存在。