TY - JOUR
T1 - Definable v -topologies, Henselianity and NIP
AU - Halevi, Yatir
AU - Hasson, Assaf
AU - Jahnke, Franziska
N1 - Funding Information:
First author was partially supported by the European Research Council Grant No. 338821, by ISF Grant No. 181/16 and ISF Grant No. 1382/15. Second author was supported by ISF Grant No. 181/16.
Funding Information:
Third author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044– 390685587, Mathematics Münster: Dynamics–Geometry–Structure and the CRC 878, as well as a Fellowship by the Daimler and Benz Foundation.
Publisher Copyright:
© 2020 World Scientific Publishing Company.
PY - 2019/11/15
Y1 - 2019/11/15
N2 - We initiate the study of definable V-topologies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K,v1,v2) is a bi-valued NIP field with v1 henselian (respectively, t-henselian), then v1 and v2 are comparable (respectively, dependent). As a consequence, Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.
AB - We initiate the study of definable V-topologies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K,v1,v2) is a bi-valued NIP field with v1 henselian (respectively, t-henselian), then v1 and v2 are comparable (respectively, dependent). As a consequence, Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.
KW - NIP valued fields
KW - Shelah conjecture
KW - definable V -topologies
KW - henselianity conjecture
UR - http://www.scopus.com/inward/record.url?scp=85075189441&partnerID=8YFLogxK
U2 - 10.1142/S0219061320500087
DO - 10.1142/S0219061320500087
M3 - Article
AN - SCOPUS:85075189441
SN - 0219-0613
VL - 20
JO - Journal of Mathematical Logic
JF - Journal of Mathematical Logic
IS - 2
M1 - 2050008
ER -