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.
Publisher Copyright:
© 2018 International Press of Boston Inc. All rights reserved.
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.
UR - http://www.scopus.com/inward/record.url?scp=85057117300&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85057117300&partnerID=8YFLogxK
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 -