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We extend results of Denef, Zahidi, Demeyer and the second author to show thefollowing.  (1) Rational integers have a single-fold Diophantine definition over the ringof integral functions of any function field of characteristic 0.  (2) Every c.e. set of integers has a finite-fold Diophantine definition overthe ring of integral functions of any function field of characteristic $0$.  (3) All c.e. subsets of polynomial rings over totally real number fields havefinite-fold Diophantine definitions. (These are the first examples of infiniterings with this property.)  (4) If $k$ is algebraic over $\Q$ and is embeddable into a finite extensionof $\Q_p$ for odd $p$, and $K$ is a one-variable function field over $k$, thenthe valuation ring of any function field valuation of $K$ has a Diophantinedefinition over $K$.  (5) If $k$ is algebraic over $\Q$ and is embeddable into $\R$, and $K$ is afunction 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 $O_{K,S}$ areexistentially definable. (Here $O_{K,S}$ is the ring of $S$-integers or a ringof integral functions.)

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