TY - JOUR

T1 - Universally and existentially definable subsets of global fields

AU - Eisenträger, Kirsten

AU - Morrison, Travis

N1 - Funding Information:
The first author was partially supported by National Science Foundation grant DMS-1056703. The second author was partially supported by National Science Foundation grants DMS-1056703 and CNS-1617802.

PY - 2018

Y1 - 2018

N2 - We show that rings of S-integers of a global function field K of odd characteristic are first-order universally definable in K. This extends work of Koenigsmann and Park who showed the same for ℤ in ℚ and the ring of integers in a number field, respectively. We also give another proof of a theorem of Poonen and show that the set of non-squares in a global field of characteristic ≠ 2 is diophantine. Finally, we show that the set of pairs (x, y) ∈ K˟ x K˟ such that x is not a norm in K(√y) is diophantine over K for any global field K of characteristic ≠ 2.

AB - We show that rings of S-integers of a global function field K of odd characteristic are first-order universally definable in K. This extends work of Koenigsmann and Park who showed the same for ℤ in ℚ and the ring of integers in a number field, respectively. We also give another proof of a theorem of Poonen and show that the set of non-squares in a global field of characteristic ≠ 2 is diophantine. Finally, we show that the set of pairs (x, y) ∈ K˟ x K˟ such that x is not a norm in K(√y) is diophantine over K for any global field K of characteristic ≠ 2.

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U2 - 10.4310/mrl.2018.v25.n4.a6

DO - 10.4310/mrl.2018.v25.n4.a6

M3 - Article

AN - SCOPUS:85057117300

VL - 25

SP - 1173

EP - 1204

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 4

ER -