Definability and Decidability over Algebraic Extensions of Product Formula Fields

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The main goal of this project is to increase our understanding of what is definable and decidable in the language of rings. The interest in the questions of polynomial definability and decidability dates back to the time of the solution of Hilbert's Tenth Problem (HTP). At the beginning of the XX century a famous German mathematician David Hilbert asked the following question (among others): is there an algorithm that can determine whether an arbitrary polynomial equation in several variables and with integer coefficients has integer solutions? In the early 1970's, Yurii Matiyasevich, building on the work by Martin Davis, Hilary Putnam and Julia Robinson showed that the sets of integers which can be defined using polynomial equations and the sets of integers that can be listed by a computer program were the same, and thus showed that an algorithm sought by Hilbert did not exist. Matiyasevich's result immediately raised another question which proved to be even more vexing:is there an algorithm as described above but for the solutions in rational numbers? This problem is unsolved to this day.As is often the case with difficult problems in Mathematics, HTP for rational numbers as well as its sister problem, HTP for the rings of integers of number fields, generated many new questions such as Mazur's conjectures and various conjectures for elliptic curves. Many of these questions turned out to be questions of Number Theory or Algebraic Geometry, but they in turn generated quite interesting consequences in Logic. The author of this proposal believes that this line of research will continue to reveal new areas of interaction between Number Theory, Algebraic Geometry and Logic, enriching all the fields involved.The language of rings or the language of polynomial equations is widely used in almost all branches of Mathematics, sciences and social sciences, and therefore understanding what can be expressed by this language and whether we can determine algorithmically which sentences in this language are true is of importance to many areas of Mathematics and beyond.