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- W2624892887 abstract "We extend results of Denef, Zahidi, Demeyer and the second author to show the following. 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. Every c.e. set of integers has a single-fold Diophantine definition over a polynomial ring over an integral domain Z of characteristic 0. 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.) 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. 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.)" @default.
- W2624892887 created "2017-06-23" @default.
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- W2624892887 date "2022-04-01" @default.
- W2624892887 modified "2023-09-30" @default.
- W2624892887 title "On existential definitions of c.e. subsets of rings of functions of characteristic 0" @default.
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- W2624892887 doi "https://doi.org/10.1016/j.apal.2021.103076" @default.
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