Uniform Definability and Undecidability in Classes of Structures

TL;DR

This paper introduces a concept of uniform encodability of theories, develops related tools, and applies them to establish broad undecidability results across large classes of structures, including function fields and polynomial rings.

Contribution

It defines uniform encodability and uses it to prove general undecidability results applicable to extensive families of algebraic structures.

Findings

Established uniform undecidability results for classes of structures

Defined uniform equivalence relations in algebraic contexts

Developed tools for analyzing theories' encodability

Abstract

We present a concept of uniform encodability of theories and develop tools related to this concept. As an application we obtain general undecidability results which are uniform for large families of structures. In the way, we define uniformly in the characteristic the equivalence relation "$x\sim y$ if and only if $x$ is an iterate of $y$ throuh the Frobenius map, or vice versa" in large classes of function fields and in polynomial rings.