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

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.)