# On existential definitions of C.E. subsets of rings of functions of characteristic 0

## TL;DR

This paper extends Diophantine definability results to rings of functions of characteristic 0, showing that various c.e. sets and valuation rings are definable, advancing understanding of number theory and logic in algebraic structures.

## Contribution

It provides the first examples of infinite rings with finite-fold Diophantine definitions and extends definability results to valuation rings and rings of S-integers in function fields.

## Key findings

- Rational integers have a single-fold Diophantine definition over integral functions.
- Every c.e. set of integers has a finite-fold Diophantine definition.
- All c.e. subsets of polynomial rings over totally real number fields are definable.

## Abstract

We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2) Every c.e. set of integers has a finite-fold Diophantine definition over the ring of integral functions of any function field of characteristic $0$. (3) All c.e. subsets of polynomial rings over totally real number fields have finite-fold Diophantine definitions. (These are the first examples of infinite rings with this property.) (4) If $k$ is algebraic over $\Q$ and is embeddable into a finite extension of $\Q_p$ for odd $p$, and $K$ is a one-variable function field over $k$, then the valuation ring of any function field valuation of $K$ has a Diophantine definition over $K$. (5) If $k$ is algebraic over $\Q$ and is embeddable into $\R$, and $K$ is a function field over $k$, then "almost" all function field valuations of $K$ have a valuation ring Diophantine over $K$. (6) Let $K$ be a one-variable function field over a number field and let $S$ be a finite set of its primes. Then all c.e. subsets of $O_{K,S}$ are existentially definable. (Here $O_{K,S}$ is the ring of $S$-integers or a ring of integral functions.)