The general questions associated with the problems discussed in this proposal can be stated as follows: what is decidable and definable in the language of rings? The current line of research has its origins in Hilbert's Tenth Problem and its solution by Davis, Putnam, Robinson and Matiyasevich. When Yurii Matiyasevich showed that over rational integers, Diophantine sets were the same as computably enumerable sets, he showed that objects originating in two different branches of Mathematics were the same. This result initiated a search for common roots of algebra and logic, a search centered on the two most important objects: integers and rational numbers. While the methods employed in the search are for the most part algebraic or geometric, the questions asked originate in Model Theory or Computability Theory. Mazur's conjectures, their extensions and their consequences for existence of Diophantine definitions and Diophantine models are a fine example of a question in Computability Theory/ Model Theory generating a question in Algebraic Geometry which in turn produces a consequence in Computability Theory/Model Theory. Another example of interaction between the fields spurred by this line of research are the recent results by Mazur and Rubin concerning ranks of elliptic curves over number fields which were produced to answer some questions of Diophantine definability. In this proposal the author plans to continue her investigation of definability and decidability over algebraic extensions of rational numbers and rational function fields of all characteristics. In particular, we plan to explore first-order definability over infinite algebraic extensions of global fields and existential undefinability of a large class of function fields of positive characteristic.
This proposal investigates questions of the following sort: given a polynomial equation in several variables, is there an algorithm (or a computer program) that can tell us if this equation has solutions in a particular set. These kinds of questions go back to a German Mathematician David Hilbert who asked at the beginning of the XX century whether we could algorithmically decide when a polynomial equation in several variables with integer coefficients has integer solutions. It turned out that such an algorithm does not exist and the solution by Davis, Putnam, Robinson and Matiyasevich took quite a few years. Unfortunately, we still don't know whether there is an algorithm to determine existence of solutions in rational numbers (fractions) as opposed to integers. That problem turned out to be even harder than the original one. As it is often the case with difficult problems, in order to solve it or even just to understand better the difficulties, an area of Mathematics has been created. This area considers polynomials as a part of a special Mathematical language (used by many areas of Mathematics and many sciences) and tries to determine what one can say in this language and whether one can decide algorithmically whether a sentence in this language is true. The current project considers these questions in various settings such as ``big rings'' (subsets of all rational numbers (fractions)) and function fields (objects formed by considering ratios of polynomials).
