A note on Diophantine subsets of large fields

TL;DR

This paper proves that in perfect large fields, finite unions of affine translates of infinite proper subfields cannot be diophantine, extending previous results and answering a question posed by Pop.

Contribution

It generalizes a result of Fehm to perfect large fields, showing certain subsets are not diophantine, thus advancing understanding of Diophantine sets in large fields.

Findings

Finite unions of affine translates of infinite proper subfields are not diophantine in perfect large fields.

Generalizes Fehm's result to a broader class of fields.

Answers a question of Pop regarding Diophantine subsets in large fields.

Abstract

Large fields (also called ample, anti-mordellic) generalize many fields of classical interest, such as algebraically closed fields, real-closed fields, and $p$-adic fields. In this note we answer a question of Pop by generalizing a result of Fehm and prove that finite unions of affine translates of infinite proper subfields are never diophantine subsets of perfect large fields.