  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.)
