Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-29T01:10:29.921Z Has data issue: false hasContentIssue false
  • English
  • Français

HENSELIAN VALUED FIELDS AND inp-MINIMALITY

Published online by Cambridge University Press:  29 August 2019

ARTEM CHERNIKOV  and
PIERRE SIMON
ARTEM CHERNIKOV
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA90095-1555, USA E-mail: chernikov@math.ucla.edu
PIERRE SIMON
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA94720-3840, USA E-mail: simon@math.berkeley.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every ultraproduct of p-adics is inp-minimal (i.e., of burden 1). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.

Keywords

03C4503C6012L1212J25valued fieldshenselianityinp-minimalityNTP2
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1510 - 1526
Copyright
Copyright © The Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adler, H., Strong theories, burden, and weight, preprint, 2007.Google Scholar
Ax, J., On the undecidability of power series fields. Proceedings of the American Mathematical Society, vol. 16 (1965), no. 4, p. 846.Google Scholar
Bélair, L., Types dans les corps valués munis d’applications coefficients. Illinois Journal of Mathematics, vol. 43 (1999), no. 2, pp. 410–425.CrossRefGoogle Scholar
Chernikov, A., Indiscernible sequences and arrays in valued fields. RIMS Kokyuroku, vol. 1718 (2010), pp. 127–131.Google Scholar
Chernikov, A., Theories without the tree property of the second kind. Annals of Pure and Applied Logic, vol. 165 (2014), no. 2, pp. 695–723.CrossRefGoogle Scholar
Chernikov, A. and Hils, M., Valued difference fields and NTP2. Israel Journal of Mathematics, vol. 204 (2014), no. 1, pp. 299–327.CrossRefGoogle Scholar
Chernikov, A., Kaplan, I., and Simon, P., Groups and fields with NTP2. Proceedings of the American Mathematical Society, vol. 143 (2015), no. 1, pp. 395–406.CrossRefGoogle Scholar
Delon, F., Types sur c((x)), Study Group on Stable Theories, Second Year: 1978/79 (Poizat, B., editor), Exp. No. 5, Secrétariat Math., Paris, 1981.Google Scholar
Dolich, A. and Goodrick, J., Strong theories of ordered Abelian groups. Fundamenta Mathematicae, vol. 236 (2017), pp. 269–296.CrossRefGoogle Scholar
Dolich, A., Goodrick, J., and Lippel, D., Dp-minimality: Basic facts and examples. Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 267–288.CrossRefGoogle Scholar
Flenner, J., Relative decidability and definability in Henselian valued fields, this Journal, vol. 76 (2011), no. 04, pp. 1240–1260.Google Scholar
Hempel, N. and Onshuus, A., Groups in NTP2. Israel Journal of Mathematics, vol. 217 (2017), no. 1, pp. 355–370.CrossRefGoogle Scholar
Johnson, W. A., Fun with fields, Ph.D. thesis, UC Berkeley, 2016.Google Scholar
Johnson, W., The canonical topology on dp-minimal fields. Journal of Mathematical Logic, vol. 18 (2018), no. 02, p. 1850007.CrossRefGoogle Scholar
Montenegro, S., Pseudo real closed fields, pseudo p-adically closed fields and NTP2.. Annals of Pure and Applied Logic, vol. 168 (2017), no. 1, pp. 191–232.CrossRefGoogle Scholar
Shelah, S., Classification Theory and the Number of Nonisomorphic Models, second ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland, Amsterdam, 1990.Google Scholar
Shelah, S., Strongly dependent theories. Israel Journal of Mathematics, vol. 204 (2014), no. 1, pp. 1–83.CrossRefGoogle Scholar
Simon, P., On dp-minimal ordered structures, this Journal, vol. 76 (2011), no. 2, pp. 448–460.Google Scholar
Touchard, P., Burden in henselian valued fields, preprint, 2018, arXiv:1811.08756.Google Scholar