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Artin-Schreier extensions and combinatorial complexity in henselian valued fields

Blaise Boissonneau

Abstract

We give explicit formulas witnessing IP, \IPn or TP2 in fields with Artin-Schreier extensions. We use them to control $p$-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the \NIPn context one way of Anscombe-Jahnke's classification of NIP henselian valued fields. As a corollary, we obtain that \NIPn henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

Artin-Schreier extensions and combinatorial complexity in henselian valued fields

Abstract

We give explicit formulas witnessing IP, \IPn or TP2 in fields with Artin-Schreier extensions. We use them to control -extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the \NIPn context one way of Anscombe-Jahnke's classification of NIP henselian valued fields. As a corollary, we obtain that \NIPn henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

Paper Structure

This paper contains 30 sections, 31 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Combinatorial complexity
  3. Complexity of henselian valued fields
  4. NIP fields
  5. Baldwin-Saxl's condition
  6. $\Rightarrow$:
  7. $\Leftarrow$:
  8. Artin-Schreier closure and local NIPity
  9. Lifting
  10. NIPn fields
  11. Baldwin-Saxl-Hempel's condition
  12. $\Leftarrow$:
  13. $\Rightarrow$:
  14. Artin-Schreier closure of NIPn fields
  15. Lifting
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $K$ be an infinite field of characteristic $p>0$. Then has IP$_{\!n}$ iff $K$ has an Artin-Schreier extension, and has TP2 iff it has infinitely many distinct Artin-Schreier extensions.

Figures (2)

  • Figure 1: A TP2 pattern
  • Figure 2: After finding $j_{2k-5},\!\cdots\!,j_{2k-1}$, we connect the $R_3(N)$ many points $j_{2k-5}-1,\!\cdots\!,j_{2k-5}-R_3(N)$ with edges colored as indicated; we seek either a monochromatic $N$-clique or two non-connected points that we then name $j_{2k-6}$ and $j_{2k-7}$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5: Anscombe-Jahnke, AJ-NIP
  • Definition 1.6: Standard Decomposition
  • Theorem 2.1: KSW
  • proof : Proof summary
  • Definition 2.2
  • Proposition 2.3: Baldwin-Saxl
  • ...and 50 more