Submission date: 11 June 2017

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) Let K be a one-variable function field over a number field and let p be any prime of K. Then the valuation ring of p has a Diophantine definition.
(5) 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.)

Mathematics Subject Classification: 11U05 (primary), 12L05, 03B25

Keywords and phrases:

Full text arXiv 1706.03302: pdf, ps.