Mathematical Sciences: Model Theory and Algebra

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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.