Mathematical Sciences: Model Theory and Algebra

本项目继续发展一种理论，用于处理微分方程系统解析解的可决性问题。例如，首席研究员和J. Denef先前已经证明，没有算法可以决定单个常微分方程在非负实数上是否有解析解，即使它在零附近有唯一解。项目的第一部分是非阿基米德刚性半解析集和子解析集理论的发展。（这里指的是非阿基米德、代数封闭、完全赋范域上的半解析集和亚解析集的理论。）这一发展将平行于最近对Denef和van den Dries的p进半解析集和亚解析集的处理。首席研究员已经证明了一个解析消去定理（在非阿基米德，代数封闭，完全赋范域上），这将促进理论的发展。所寻求的结果包括(a)刚性子解析集的补是子解析的；(b)每个子解析集都是子解析流形的有限不相交并；(c)刚性亚解析集和函数的非阿基米德Lojasiewicz不等式；还有一些进一步的结果。在第二部分中，主要研究者试图证明p进整数Zp的代数子集，由涉及少量单项式的多项式定义，如果它们是有限的，则在Zp中只包含几个点。在第三部分中，他将研究一个关于最大超定偏微分方程组幂级数解存在性的判定问题。

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.