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Uniform Properties of Ideals in Rings of Restricted Power Series

Part of: Diophantine equations Applications of logic to commutative algebra

Published online by Cambridge University Press:  28 June 2022

Madeline G. Barnicle
Madeline G. Barnicle*
Affiliation:
University of California, Los Angeles, USA, 2019.
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Abstract

When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $ , $\mathbb Z_{p}\langle X\rangle $ , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg.

Abstract prepared by Madeline G. Barnicle.

E-mail: barnicle@math.ucla.edu

URL: https://escholarship.org/uc/item/6t02q9s4

Keywords

model theoryp-adic numbersdefinabilityprime idealsradical idealsWeierstrass Division

MSC classification

Primary: 11D88: $p$-adic and power series fields
Secondary: 13L05: Applications of logic to commutative algebra
Type
Thesis Abstracts
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 2 , June 2022 , pp. 258
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

The Association for Symbolic Logic publishes abstracts of recent PhD theses in logic. The aim of this activity is to publish abstracts for the majority of recent PhD theses in logic world wide and submitted abstracts will therefore only be edited to ensure that they fall within the general area of logic and are appropriate in terms of length and content. This section will provide a permanent publicly accessible overview of theses in logic and thus make up for the lack of central repository for the theses themselves. The Thesis Abstracts Section is edited by Christian Rosendal. Any abstract should formally be submitted by the thesis advisor though it is expected to usually be prepared by the candidate. For detailed instructions for preparation and submission, including the required TeX template, please consult the link below. http://aslonline.org/LogicThesisAbstracts.html

Footnotes

Supervised by Matthias Aschenbrenner.