Mathematical Sciences: Model Theory and Algebra

该项目继续发展一种理论， 处理关于解析解的可判定性问题 微分方程组 例如，校长 研究人员和J. Denef先前已经表明， 一种判定单个常微分方程 在非负真实的数上有解析解，甚至 因为它有唯一的接近于零的解。 第1部分 项目是一个非阿基米德刚性理论的发展 半解析集和次解析集 (By这意味着 环上半解析与次解析集理论 nonarchimedean，algebraically closed，complete normed field.） 这一发展将与最近对p-adic的处理并行 Denef和货车den Dries的半解析和次解析集。 首席研究员已经证明了一个分析消除 定理（在非阿基米德，代数闭，完全 这将有助于发展中国家的发展。 理论 所寻求的结果包括：（a） 刚性子解析集是子解析的;（B）每个子解析集 集合是次解析流形的有限不交并;（c） 刚性次解析的非阿基米德Lojasiewicz不等式 集合和函数;以及一些进一步的结果。 在第二部分中， 首席研究员试图表明，代数子集的 p进整数Zp，由涉及较少的多项式定义 单项式，如果它们是有限的，只包含Zp中的几个点。 在 第三部分，他将研究一个关于 最大超定方程幂级数解的存在性 偏微分方程组

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.