Non-Diophantine sets in rings of functions

TL;DR

This paper investigates the Diophantine properties of certain sets and relations in polynomial rings and fields of rational functions, showing many are not Diophantine, thus advancing understanding of their algebraic complexity.

Contribution

It proves that several natural sets and relations in polynomial rings and rational function fields are not Diophantine, providing new insights into their algebraic structure.

Findings

Several natural sets in polynomial rings are not Diophantine.

Many relations over rational function fields are shown to be non-Diophantine.

Progress towards classifying Diophantine sets in these algebraic structures.

Abstract

Except for a limited number of cases, a complete classification of the Diophantine sets of polynomial rings and fields of rational functions seems out of reach at present. We contribute to this problem by proving that several natural sets and relations over these structures are not Diophantine.