On Existential Definitions of C.e. Subsets of Rings of Functions of Characteristic 0
نویسندگان
چکیده
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 OK,S are existentially definable. (Here OK,S is the ring of S -integers or a ring of integral functions.)
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