On existential definitions of c.e. subsets of rings of functions of characteristic 0

We extend results of Denef, Zahidi, Demeyer and the second author to show the following.(1)Every c.e. set of integers has 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 single-fold Diophantine definition over a polynomial ring over an integral domain Z of characteristic 0.(3)All c.e. subsets of polynomial rings over rings of totally real integers have finite-fold Diophantine definitions. (These are the first examples of infinite rings with this property.)(4)Let K be a one-variable function field over a field of constants k, and let p be any prime of K. If k is algebraic over Q and for some odd prime p embeddable into a finite extension of Qp, then the valuation ring of p has a Diophantine definition over K. If k is embeddable into a real field, then valuation rings are existentially definable for “almost all” primes.(5)Let K be a one-variable function field over a number field and let S be a finite non-empty set of its primes. Then all c.e. subsets of OK,S are Diophantine over OK,S. (Here OK,S is the ring of S-integers or a ring of integral functions.)